Although the Kling-Gupta efficiency ($\mathrm{KGE}$) is widely adopted for model evaluation in hydrology, its properties as a statistical estimator remain unexplored. Investigating these properties is necessary because parameter estimation and forecast evaluation are inherently linked. To address this, we formalize the negatively oriented Kling-Gupta loss $L_\mathrm{KG} = (1 - \mathrm{KGE})^2$ within an extremum estimation framework (equivalent to maximizing $\mathrm{KGE}$) and analyze its behavior in multiple linear regression. We establish explicit formulas for the parameter estimates, showing that Kling-Gupta linear regression scales the ordinary least squares (OLS) coefficient vector by a variance-inflation factor governed by the sample variances and covariances of the predictors and the response. We show that Kling-Gupta linear regression predictions replicate the sample variance of the response on the training set, in contrast to the variance reduction inherent to OLS, while both estimators maintain the sample mean of the observations and achieve the same sample correlation between the predictions and the response. We show analytically that no single estimator can simultaneously maximize both the Nash-Sutcliffe efficiency $\mathrm{NSE}$ and $\mathrm{KGE}$: the OLS estimator attains the maximum possible $\mathrm{NSE}$ but not the maximum $\mathrm{KGE}$, while the Kling-Gupta estimator maximizes $\mathrm{KGE}$ at the cost of $\mathrm{NSE}$. We prove the almost sure convergence of the Kling-Gupta estimator to well-defined population limits and express those limits algebraically. Furthermore, we evaluate the training and test set performance metrics for both estimators, demonstrating that for each estimator the metrics on the training set and on an independent test set converge asymptotically to identical limits (though the limits differ between OLS and Kling-Gupta regression).

Knowledge transfer across data sources holds great promise for improving the estimation of target population parameters by leveraging the growing availability of data from different sources. However, the effectiveness of knowledge transfer is often challenged by the complex and pervasive heterogeneity between data sources and the lack of access to individual-level data. This paper proposes the divide-and-shrink (dShrink) method, a transfer estimation method that estimates target population parameters in a closed form using summary statistics from a target population and some external source populations while accounting for population heterogeneity. The dShrink estimator is guaranteed to outperform the estimator based solely on the target population in terms of expected quadratic error under arbitrary population heterogeneity. The gain can be substantial when the target and source populations are similar, or the underlying true parameter values are near zero. Notably, dShrink is model-free, requires no user-specified tuning parameters, robust to various types of heterogeneity between data sources, and applies to a broad range of parameter estimation problems. dShrink remains effective even when the covariance matrix is not accessible for the external summary statistics and offers flexibility in incorporating side information and summary statistics from multiple source populations. Simulations and real data analyses demonstrate the superior performance of the dShrink estimator and its potential as a robust tool for transfer estimation.

The strong likelihood principle (SLP) is conventionally derived from the sufficiency principle and a conditionality principle in an argument due to Birnbaum, and much of the literature contests whether the derivation is sound. We take a different approach. We ask what the SLP says when its terms are read carefully, and argue that the principle as ordinarily stated reflects a confusion about the domain of the likelihood function. The likelihood is naturally defined as a function on a family of distributions $M$, not on a parameter space, and once it is so defined the SLP collapses into its weak counterpart, the weak likelihood principle. The diagnosis is illustrated by analogy with monetary value, developed concretely through a comparison of the binomial and negative binomial families that share a parameter, and connected to the geometric structure of $M$ through the Fisher information metric. The same standardization emerges from a statistical argument about comparing measurements across populations and from a geometric argument about manifold distance; this convergence supplies the positive content of the weak likelihood principle.

There is growing interest in constructing conformal prediction sets that provide approximate or asymptotic conditional coverage guarantees, capturing local data heterogeneity. However, methods like localized conformal prediction (LCP) may face challenges in ensuring reliable prediction sets in regions with sparse calibration data. This paper introduces Enhanced Localized Conformal Prediction (ELCP), a novel approach that incorporates auxiliary data to refine localized prediction sets while preserving finite-sample marginal coverage guarantees. By utilizing a density-ratio-weighted kernel estimator, ELCP seamlessly integrates auxiliary and calibration data, accommodating potential distributional shifts and improving the local reliability of prediction sets. Theoretical analysis confirms that ELCP maintains marginal coverage and enhances asymptotic test-conditional coverage. Simulation results demonstrate its superior local coverage and smaller prediction sets compared to standard LCP, highlighting its effectiveness in settings with limited calibration data but available auxiliary information from related tasks.

Control-variate and polynomial-maximization (PMM) estimators are optimized at a single fixed distribution, yet they are increasingly proposed to strengthen hypothesis tests, which decide between two regions of a parameter family. We give a closed-form criterion for when this transfer succeeds. For an H0-centered augmentation of a target moment statistic with null-optimized weight vector K0, the alternative-side expectation equals the target plus K0^T mu_a,H1, where mu_a,H1 is the alternative-side mean of the augmenting basis. Null-variance reduction therefore transfers without bias only under the orthogonality condition K0^T mu_a,H1 = 0; requiring each augmenting function to remain mean-zero is sufficient but not necessary. We instantiate the criterion on the recently proposed Wiedermann-Shi third-order cumulant test for measurement-error independence. A second-order PMM correction is unbiased and lower-variance under the null (relative efficiency >= 1 in all 36 conditions; aggregated mean ARE values 1.23-5.16; Type-I 0.04-0.09), yet provably inconsistent under the alternative: the antisymmetric polynomial auxiliaries acquire nonzero means, attenuating the target by a closed-form factor and costing 7-52 percentage points of power, worst where the test is strongest and worsening under heavy tails. A fourth-order variant reduces variance (ratio 1.127) but fails a nuisance guard (rejection 0.295 versus 0.10). We derive a reusable alternative-consistency acceptance gate for variance-reduced test statistics.

Ordinary differential equation models are widely used to understand and forecast complex dynamical systems, but their predictive value depends on reliable parameter estimation. Structural identifiability assesses whether parameters can be uniquely recovered from ideal observations, whereas practical identifiability depends on finite, noisy and partially observed data. We introduce the Practical Identifiability Index (PII), a marginal uncertainty-width metric based on the logarithmic span of confidence intervals. Expressed on an order-of-magnitude scale, the PII summarises how tightly individual positive-valued parameters are constrained by available observations, enabling comparison across parameters, models, error structures and observation designs. The PII is intended as a complementary diagnostic, not a standalone identifiability test, and should be interpreted alongside coverage, profile likelihoods, posterior summaries, sensitivity analysis or structural identifiability results. Using parametric bootstrap experiments across growth and compartmental epidemic models, we identify consistent principles: uncertainty decreases as calibration windows become more informative, increases with observation noise and parameter coupling, and remains high for latent or indirectly observed processes. Parameters governing early observable dynamics become constrained sooner, while additional observables can improve constraint for latent progression and recovery parameters. The PII provides a simple, reportable summary of marginal parameter uncertainty for dynamical modelling.

Data-adaptive two-sample testing assesses if two samples come from the same distribution, using a discrepancy learned from the data (e.g., via kernel-based feature representations). Such methods typically rely on data splitting to decouple learning from testing and control type I error. However, this paradigm is ill-suited to few-shot settings with severe sample-size imbalance: abundant reference samples are available, while only a handful of query samples arrive. In this paper, we show how this imbalance can be leveraged constructively. Using abundant reference data, we learn reference-dependent representations that summarize salient structure of the reference distribution and provide informative signals for detecting departures. We incorporate a collection of representation families that capture both global and local structure, and adaptively weight them using only reference samples via an uncertainty-guided principle. Theoretically, we establish permutation-based type I error control and show consistency of the aggregated test: as the sample sizes grow, the test power converges to one whenever the representation set contains at least one consistent representation. Empirically, our aggregation achieves strong performance across a range of benchmarks while retaining type I error control.

Phylogenetic networks are graphs inferred from molecular sequence data that represent ancestral histories shaped by reticulate processes such as recombination, hybridization, and horizontal gene transfer. We introduce a family of distance metrics for rooted, ranked, unlabeled phylogenetic networks, extending a previously developed distance for ranked trees. Our approach relies on a bijective triangular matrix representation of phylogenetic networks that captures the temporal order of internal events, speciations, and hybridizations. Our metrics, defined as standard matrix norms, allow efficient quantitative comparisons of network topologies, timed networks and networks with differing numbers of hybridizations. Our distance can be used for both isochronous networks where all tips are sampled at one time point, and heterochronous networks where tips are allowed to be sampled at different time points. We show that our metrics capture biologically meaningful differences among evolutionary histories in both simulations and empirical posterior distributions of viral phylogenetic networks. These tools fill a methodological gap, enabling principled comparisons of ranked, unlabeled phylogenetic networks, including ancestral recombination graphs.

Characterizing directed information flow in brain networks is difficult because neural circuits are full of recurrent feedback loops. Many existing tools for directed dependence assume a directed acyclic graph (DAG) structure to resolve directional ambiguity, and therefore cannot represent these loops. We present a nonparametric, information-theoretic framework that addresses this by coupling the discrete Hodge decomposition with lead-lag mutual information, splitting the resulting edge flow into three orthogonal components: a gradient term capturing hierarchical, feed-forward relationships; a curl term isolating triangle-level feedback loops; and a harmonic term capturing cyclic flow around topological holes. This separation makes it possible to disentangle feed-forward drive from recurrent circulation, which conventional measures conflate. We further develop a permutation-based hypothesis-testing layer that identifies nodes and triangular motifs whose information-flow signatures change significantly between conditions. We validate the framework on simulations with known ground-truth structure and apply it to local field potential recordings from a rodent model of focal ischemic stroke. In three of four animals, we find a post-stroke shift toward hierarchical, source-driven propagation at the expense of recurrent feedback, while the fourth shows no significant change.

To characterize the US airline profit cycles from 1995 to 2020, the authors of Renold et al. (2023) combine k-means clustering, principal component analysis, and system dynamic modelling. We replicate their clustering experiment in three spaces -- the original 7-dimensional raw-variable space, a 3-dimensional PC score space, and a 4-dimensional PC score space using their dataset gratefully included in the paper. We show that the six-cluster taxonomy is geometrically robust: k-means in 3-PC space produces bit-for-bit identical cluster assignments relative to 7D raw space. As a nonlinearity check we apply kernel PCA under six kernels spanning three families plus a linear baseline. All six kernels preserve the six-cluster assignment in 2D. A 1D diagnostic tightens this: the linear kernel conflates the COVID year C_3 with the peak-profit cluster C_0, whereas all five non-baseline kernels shift C_3 to overlap only the post-financial-crisis cluster C_5. Agreement across the kernel families confirms an intrinsically linear manifold with no hidden curvature. The silhouette criterion reveals that the dataset structurally supports only three clusters, not six. Collinearity in the raw 7D space suppresses the silhouette signal that would otherwise identify k=3 as the structurally motivated choice.

Most statistical and machine learning methods for directed interactions focus on pairwise effects among variables. Even existing cyclic models represent feedback primarily through node-level dependencies, making large-scale recurrent organization difficult to estimate and compare. This limitation is particularly acute in biological and neural systems, where interactions are highly recurrent and involve many overlapping cycles. We introduce a variational framework for statistical inference on cyclic interactions. Directed interactions are represented as edge flows on a simplicial complex and evolved under an energy-minimizing dynamical system. The resulting dynamics separate transient interaction components from persistent harmonic flows, yielding a low-dimensional cycle space that captures stable recurrent organization. Rather than enumerating individual cycles, the proposed framework represents cyclic interactions as elements of a Hilbert space, enabling projection, averaging, comparison, and population-level statistical inference. We establish theoretical properties of the harmonic projection, including characterization of the cycle space, variance reduction, and population inference. Simulations demonstrate substantially improved recovery of cyclic structure in dense recurrent systems compared with existing directed-interaction methods. Applied to resting-state fMRI from 400 human subjects, the framework reveals reproducible large-scale cyclic organization that is not detectable through edgewise averaging. These results provide a scalable statistical framework for studying recurrent interactions in high-dimensional dynamical systems.

The main goal in regression modelling consists in approximating the conditional mean of a response given a set of features. A regression function is said to be calibrated if the resulting mean estimates match the true conditional means for almost every set of features. Aiming for calibration seems not achievable in practice as one typically deals with finite samples of noisy observations. A weaker notion of calibration is auto-calibration, and it means that the expectation of responses being given the same mean estimate matches this estimate. This notion is important, e.g., in insurance pricing as it ensures no cross-subsidization between different price cohorts. In this paper, we show that boosting trees can be used to test necessary conditions for calibration and auto-calibration, respectively. The practical relevance of our approach is supported by a numerical example, in which the proposed tests prove to be very powerful on a large insurance dataset.

This study investigates a statistical property of Lagrange multipliers in constrained Maximum Likelihood Estimation (MLE) and Least Squares (LS) problems from the perspective of numerical optimization. Building on large-sample theory, we show that the associated Lagrange multipliers converge to zero as the sample size increases, provided the distribution is correctly specified in MLE or the residuals are normally distributed in LS. Although this asymptotic behavior has long been recognized in statistics, it has received little explicit attention in numerical optimization and has rarely been exploited in algorithmic design. Importantly, the insight extends beyond classical low-dimensional settings: even in modern high-dimensional applications, such as deep learning, where the number of parameters may exceed the sample size, the same reasoning applies provided the generalization performance is good. This observation has two main implications. First, many constrained optimization algorithms, including the Augmented Lagrangian Method, Sequential Quadratic Programming, and Interior Point methods, require initial values for the multipliers, and choosing zero is statistically justified. Numerical experiments for constrained regressions and dynamic discrete choice model estimations support this implication by showing that initializing multipliers at zero usually lead to stable and efficient performance. Second, penalty-based approaches that convert constrained problems into unconstrained ones can perform well when the true multipliers are small. This helps explain why penalty-based methods often perform well in practice.

We study component recovery and mixing-matrix estimation from unlabeled finite mixtures whose observable distributions share the same latent components but have unknown mixing weights. The main identifying signal is marginal independence: each component is assumed to be independent on at least one coordinate pair, but no labels, clean component samples, or mixing weights are observed. We first prove a structural result for product components: under linear independence of the univariate marginals, any independent affine combination of the components must coincide with a single component. We then extend this principle to observable mixtures and show that, under full-rank and no-cancellation conditions, marginally independent affine combinations recover the corresponding latent components. When every component is independent on some coordinate pair, all components are identifiable, and the mixing matrix is recoverable under the stated completion conditions. Finally, we propose a Product-Marginal Maximum Mean Discrepancy (PM-MMD) estimator over affine combinations of the observable mixtures and prove uniform convergence and stability under approximate marginal independence. This framework also separates the empirical roles of the assumptions: irreducibility is, in general, not directly testable from the unlabeled mixtures alone, whereas marginal independence yields a candidate-level diagnostic through held-out PM-MMD. Controlled and flow-cytometry experiments show when marginal independence provides a useful recovery signal. In the reported multi-component comparisons, condition-aware representative selection stabilizes PM-MMD and improves recovery relative to clustering, factorization, and pairwise mixture-proportion baselines using the same unlabeled mixtures.

Sensitivity analysis is an important component of simulation-based decision support because it helps analysts determine which inputs most strongly influence model outcomes under uncertainty. This paper organizes the broad sensitivity analysis literature into a coherent framework for use in complex simulation settings, with particular attention to military applications. We review major classes of methods, including local and global approaches, variance-based techniques, screening methods, derivative-based methods, and uncertainty quantification tools, and relate them to common analytical objectives such as factor prioritization, factor fixing, variance reduction, and factor mapping. The paper also discusses sensitivity auditing as a complementary perspective that emphasizes transparency, assumption tracking, and responsible use of models in decision-relevant settings.

Relational Event Models (REMs) provide a rigorous framework for analyzing dyadic interactions observed in continuous time, capturing history-dependent dynamics such as triadic closure and reciprocity. Framing REMs through the lens of counting processes embeds the model in a rich theoretical foundation, facilitating its mathematical analysis. While Maximum Likelihood Estimation (MLE) is standard practice for estimating these models, the underlying statistical guarantees rely on specific asymptotic regimes, namely, whether the network size (n), the observational period (T), or both approach infinity. We review the theoretical foundations of such counting-process-based models, formalizing the core assumptions required to achieve asymptotic normality across these different limits. With a specific focus on Cox-type multiplicative models, we detail the circumstances under which these assumptions hold. Supported by simulation studies, we illustrate how structural modeling choices, including temporal windowing and logarithmic transformations, affect empirical coverage and estimator convergence. We thereby derive several guiding principles for specifying such models in realistic contexts, bridging theory and practice.

Gaussian processes with stationary kernels on bounded domains exhibit inflated posterior variance near the boundary. Despite being a long-recognized artifact in geostatistics and a source of over-exploration in Bayesian optimization, the causes and effects of boundary-induced acquisition bias are underexplored. We trace the root cause to a simple geometric mechanism: the truncation of the kernel correlation neighborhood at the domain boundary creates an observation-independent distortion that worsens with dimensionality. We show how this distortion manifests across three acquisition classes: variance maximization concentrates selections at the corners, whereas negative integrated posterior variance and expected predictive information gain move selections inward to axis-aligned interior shells. These patterns arise without reference to any objective function, meaning that acquisition behavior can be dominated by kernel geometry rather than the desired task-specific uncertainty. To quantify this, we introduce a function-free selection-profile diagnostic for arbitrary acquisitions, kernels, and bounded-domain geometries.

Multidimensional Gaussian Process (multi-GP) regression is widely used to disentangle stellar and planetary signals in radial velocities (RVs) by jointly modelling ancillary activity indicators. However, identifying the combination of indicators that best constrains the stellar signal in the RVs is non-trivial, as classical model comparison methods are not directly applicable when multi-GPs involve different time series combinations. In this work, we present an information criterion to compare multi-GP models based on their ability to explain the RV component, $\mathrm{MGIC}_{\rm rv}$. This metric combines the conditional RV likelihood with an effective parameter count that accounts for the regularisation imposed by the multi-GP model on the RV component. We demonstrate that $\mathrm{MGIC}_{\rm rv}$ provides a quantitative and robust framework for multi-GP model comparison, identifying the activity indicators that most effectively constrain the RV signal. Although developed in the context of RV analysis, the proposed criterion is general and applicable to multi-GP problems in which the inference focuses on a specific observable.

We consider the problem of determining feasible systems from a finite set of simulated alternatives with respect to probability constraints, where the observations from stochastic simulations are Bernoulli distributed. Most statistically valid procedures for feasibility determination focus on constraints on the means of normally distributed observations. Although these procedures can be adapted to Bernoulli-distributed data by treating batch means as basic observations, achieving approximate normality often requires a large batch size, potentially leading to the unnecessary waste of observations in reaching a decision. This paper proposes a procedure that utilizes the Bernoulli-distributed observations directly to determine feasibility. In addition, we incorporate subjective constraints, allowing for multiple thresholds for each constraint. We demonstrate that our proposed procedure is statistically valid and that it outperforms an existing feasibility determination procedure for subjective constraints originally developed for normally distributed observations. Furthermore, we propose two heuristic feasibility check approaches for thresholds that are sequentially added by decision makers, allowing thresholds to be tightened when many systems are feasible or relaxed when no feasible system exists. We show by experiments that the proposed procedures can efficiently provide feasibility decisions to systems with respect to all thresholds considered.

This paper proposes a novel method to estimate the rate parameter of the Poisson distribution. The proposed method employs the Cramer-von Mises type optimization which has been commonly used in estimating parameters of continuous distributions. Upon obtaining the estimator through the proposed method, its desirable properties such as asymptotic distribution and robustness are rigorously investigated. Simulation studies serve to demonstrate that the proposed method compares favorably with other well-celebrated methods including the maximum likelihood method.

Kunchenko's method of polynomial maximization provides a semiparametric apparatus for parameter estimation under non-Gaussian errors, but its classical power basis relies on finite higher-order integer moments. This paper introduces the Parametrically Adaptive Transition Polynomial (PATP), a signed-parity fractional-power family controlled by a continuous parameter alpha in [0,1]. The quadratic exponent map p_i(alpha) connects the fractal regime p_i(0)=1/i, the degenerate linear point p_i(1/2)=1, and the signed-parity integer-power regime p_i(1)=i. For the degree-S=2 case we derive a closed-form variance-reduction coefficient g_2(alpha) in terms of signed and absolute fractional moments, identify the singular behavior at alpha=1/2, and state the moment and regularity conditions under which the formula is meaningful. The construction should be read as a Form-B PATP analogue within Kunchenko's generalized apparatus, not as an exact recovery of the canonical even-power PMM basis at alpha=1. Numerical illustrations on canonical distributions are used to examine the finite-sample behavior of the signed-parity estimator and to mark the boundary of applicability for extremely heavy-tailed cases such as Cauchy.

High-fidelity (HF) data are often expensive to collect and therefore scarce, making conditional quantiles difficult to estimate accurately. We propose a two-stage, model-agnostic method for multi-fidelity quantile regression. The central idea is a local quantile link: at each covariate value, the HF quantile is represented as a low-fidelity (LF) quantile evaluated at a covariate-dependent level. This reformulation reduces the problem to estimating the level function, which can be smoother than the HF quantile itself when the LF and HF conditional distributions have similar shapes. We also study the complementary regime in which this advantage weakens and introduce a correction step to improve robustness. Our theory characterizes when the proposed estimator converges faster than direct quantile regression using HF data alone and when the correction step provides further improvement. Experiments on synthetic and real data show that our method yields more accurate quantile estimates and tighter conformal prediction intervals.

Maximum likelihood prediction (MLP) is a core task at the heart of modern large language models. Here, we study a quantum version of this task for a simplified data model consisting of independent and identically distributed samples, as a first step. The quantum maximum likelihood predictor is obtained by embedding of empirical probability distributions into quantum states and performing a minimization of quantum relative entropy over a given class of states. We provide an interpretation of this predictor in terms of quantum reverse information projection and quantum Pythagorean theorem when the class of quantum models is sufficiently expressive. We further derive non-asymptotic performance guarantees in terms of convergence rates and concentration inequalities, both in trace norm and quantum relative entropy. Our approach provides a unified framework to handle MLP within both classical and quantum LLMs.

Identifying brain regions that exhibit altered functional connectivity across cognitive or emotional states is a key problem in neuroscience. Existing methods, such as edge-wise testing, seed-based psychophysiological interaction (PPI) analysis, or correlation network comparison, typically suffer from low statistical power, arbitrary thresholding, and limited ability to capture distributed or nonlinear dependence patterns. We propose SpARCD (Spectral Analysis of Revealing Connectivity Differences), a novel statistical framework for detecting differences in brain connectivity between two experimental conditions. SpARCD leverages distance correlation, a dependence measure sensitive to both linear and nonlinear associations, to construct a weighted graph for each condition. It then constructs a differential operator via spectral filtering and uncovers connectivity changes by computing its leading eigenvectors. Inference is achieved via a permutation-based testing scheme that yields interpretable, region-level significance maps. Extensive simulation studies demonstrate that SpARCD achieves superior power relative to conventional edge-wise or univariate approaches, particularly in the presence of complex dependency structures. Application to fMRI data from 113 early PTSD patients performing an emotional face-matching task reveals distinct networks associated with emotional reactivity and regulatory processes. Overall, SpARCD provides a statistically rigorous and computationally efficient framework for comparing high-dimensional connectivity structures, with broad applicability to neuroimaging and other network-based scientific domains.

Functional ANOVA provides a nonparametric modeling framework for multivariate covariates, enabling flexible estimation and interpretation of effect functions such as main effects and interaction effects. However, effect-wise inference in such models remains challenging. Existing methods focus primarily on inference for entire functions rather than individual effects. Methods addressing effect-wise inference face substantial limitations: the inability to accommodate interactions, a lack of rigorous theoretical foundations, or restriction to pointwise inference. To address these limitations, we develop a unified framework for effect-wise inference in smoothing spline ANOVA on a subspace of tensor product Sobolev space. For each effect function, we establish rates of convergence, pointwise confidence intervals, and a Wald-type test for whether the effect is zero, with power achieving the minimax distinguishable rate up to a logarithmic factor. Main effects achieve the optimal univariate rates, and interactions achieve optimal rates up to logarithmic factors. The theoretical foundation relies on an orthogonality decomposition of effect subspaces, which enables the extension of the functional Bahadur representation framework to effect-wise inference in smoothing spline ANOVA with interactions. Simulation studies and real-data application to the Colorado temperature dataset demonstrate superior performance compared to existing methods.

Our objective is to construct well-calibrated prediction sets for a time-to-event outcome subject to right-censoring with guaranteed coverage. Inspired by modern conformal inference, our approach avoids the need for a well-specified parametric or semiparametric survival model. Unlike existing conformal methods for survival data, which assume Type-I censoring with fully observed censoring times, we consider the more common right-censoring setting in which only the censoring time or only the event time is observed, whichever comes first. Under a standard conditional independence censoring condition, we propose and analyze several lower prediction bounds for the survival time of a future observation, including inverse-probability-of-censoring weighting, and its augmented version based on the semiparametric efficient influence function for the relevant marginal quantile of the outcome accounting for dependent censoring. We formally establish asymptotic coverage guarantees of the proposed methods, and demonstrate both theoretically and through empirical experiments, that the augmented approach substantially improves efficiency over all other proposed methods. Specifically, its coverage error bound is doubly robust, and therefore of second order, thus ensuring that it is asymptotically negligible relative to the coverage error of the other methods.

Drift diffusion models (DDMs) have found widespread use in computational neuroscience, cognitive science, mathematical psychology as well as other fields. They model evidence accumulation in simple decision tasks as a stochastic process drifting towards decision barriers. In models where the drift is both time-varying within a trial and variable across trials, the high computational cost for accurate likelihood evaluation has often led to the use of a computationally convenient surrogate for parameter inference, the time-averaged drift approximation (TADA). In each trial, TADA assumes that the time-varying drift rate can be replaced by its temporal average throughout the trial. This approach enables fast parameter inference using analytical likelihood formulas for DDMs with constant drift. In this work, we show that such an estimator is inconsistent: it does not converge to the true drift, posing a risk of biasing scientific conclusions when parameter estimates are obtained by TADA and similar approximations. We provide an elementary proof of this inconsistency in what is perhaps the simplest possible setting: a Brownian motion with piecewise constant drift hitting a one-sided upper boundary. Furthermore, numerical examples based on an attentional DDM (aDDM) show that using TADA leads to systematic misestimation of attentional effects in decision making and can lead to false conclusions in scientific hypothesis testing.

With the emergence of dynamic multiplex networks, corresponding to graphs where multiple types of edges evolve over time, a key inferential task is to determine whether the layers associated with different edge types differ in their connectivity. In this work, we introduce a hypothesis testing framework, under a latent space network model, for assessing whether the layers share a common latent representation. The method we propose extends previous literature related to the problem of pairwise testing for random graphs and enables global testing of differences between layers in multiplex graphs. While we introduce the method as a test for differences between layers, it can easily be adapted to test for differences between time points. We construct a test statistic based on a spectral embedding of an unfolded representation of the graph adjacency matrices and demonstrate its ability to detect differences across layers in the asymptotic regime where the number of nodes in each graph tends to infinity. The finite-sample properties of the test are empirically demonstrated by assessing its performance on both simulated data and a biological dataset describing the neural activity of larval Drosophila.

Are two distributions close to each other with statistical significance? Distribution closeness testing (DCT) formalizes this question by testing whether the distance between a distribution pair is at least epsilon-far. Existing DCT methods mainly measure discrepancies between distribution pairs defined on discrete spaces, for example using total variation, which limits their application to complex data such as images. To extend DCT to more types of data, a natural idea is to introduce maximum mean discrepancy (MMD), a powerful measure of distributional discrepancy between complex distributions, into DCT scenarios. However, empirical results indicate that many distribution pairs can have the same MMD value despite having different norms in the same reproducing kernel Hilbert space (RKHS). These pairs may exhibit different finite-sample distinguishability and reflect different practical closeness levels, making MMD less informative for DCT. To mitigate this issue, we design a new measure of distributional discrepancy, norm-adaptive MMD (NAMMD), which scales the MMD value using the RKHS norms of distributions. Based on the asymptotic distribution of NAMMD, we propose NAMMD-based DCT to assess the closeness level of a distribution pair. Theoretically, we prove that NAMMD-based DCT has higher test power than MMD-based DCT while maintaining bounded type-I error. This is further validated by extensive experiments on multiple types of data, including synthetic noise and real images. Our code is available at https://github.com/zhijianzhouml/NAMMD.

In official statistics, dual system estimation (DSE) is a well-known tool to estimate the size of a population. Two sources are linked, and the number of units that are missed by both sources is estimated. Often dual system estimation is carried out in each of the levels of a stratifying variable, such as region. DSE can be considered a loglinear independence model, and, with a stratifying variable, a loglinear conditional independence model. The standard approach is to estimate parameters for each level of the stratifying variable. Thus, when the number of levels of the stratifying variable is large, the number of parameters estimated is large as well. Mixed effects loglinear models, where sets of parameters involving the stratifying variable are replaced by a distribution parameterised by its mean and a variance, have also been proposed, and we investigate their properties through simulation. In our simulation studies the mixed effects loglinear model outperforms the fixed effects loglinear model although only to a small extent in terms of mean squared error. We show how mixed effects dual system estimation can be extended to multiple system estimation.

When the rate parameter of the exponential distribution is associated with predictors, then the main interest will be how to estimate the regression parameter. In this paper, we will investigate how to estimate the parameter on the regression setup of the exponential distribution. To that end, we propose a new estimator, and its asymptotic properties will be discussed.

Thousands of experiments are analyzed, and papers are published each year involving the statistical analysis of grouped data. While this area of statistics is often perceived -- somewhat naively -- as saturated, several misconceptions still affect everyday practice, and new frontiers have so far remained unexplored. Researchers must be aware of the limitations affecting their analyses and what new possibilities are at their hands. The article introduces a unifying approach to the analysis of divisible statistics -- that includes Pearson's $χ^2$, the likelihood ratio, and spectral statistics, as special cases -- when a statistician deals with a large number of bins/groups, thus leading to a large number of small or moderate frequencies. Performance of the tests is analyzed against the class of contiguous (local) alternatives. Perhaps the most surprising result here is that, in this `sparse' regime, most of the tests proposed in the literature can be modified to produce more powerful tests, and no single test based on a divisible statistic leads to a goodness-of-fit test. Distribution-free goodness-of-fit tests are also constructed.

Bayesian optimization (BO) is a central tool for sample-efficient design, and latent-space Bayesian optimization (LSBO) extends it to structured objects such as molecules and proteins. In parallel, tabular foundation models such as TabPFN and TabICL now achieve state-of-the-art regression performance and are increasingly used as BO surrogates. Because their Bayesian behavior is induced by large synthetic pretraining collections, the composition of this pretraining distribution is crucial. LSBO creates a distinctive mismatch: the induced map from latent code to objective value differs markedly from the regression tasks used to train current in-context models. We address this mismatch by complementing the pretraining stage of tabular foundation model surrogates with synthetic optimization tasks defined on the latent space of a molecular VAE. The continued-pretraining objective features a regularizer that anchors the model to the original checkpoint, preserving its broad regression prior while avoiding overspecialization to the adaptation tasks. On held-out molecular optimization benchmarks, the resulting model achieves strong performance, supporting the relevance of LSBO-specific adaptation for in-context surrogates.

Multivariate regression with many correlated responses and predictors commonly violates Gaussian error assumptions due to heavy tails, outliers, and asymmetry. Gaussian procedures then lose efficiency in coefficient estimation and produce biased estimates of conditional dependence graphs. We develop a robust Bayesian framework using a scale-location mixture error distribution and horseshoe+ global-local priors on both the regression coefficients and off-diagonals of the error precision matrix, coupling sparsity in the regression map with sparsity in the residual dependence structure. Theoretical contributions include joint posterior contraction, selection consistency for both supports, a Kullback-Leibler risk bound showing the dominance of horseshoe+ over horseshoe, and bounded sensitivity, ensuring that a single large outlier has vanishing influence under t errors. Simulations across four error regimes, contamination, and varying dimensions show that our estimator matches Gaussian procedures under normality and dominates them under heavy tails and skewness. Applications to FRED-MD macroeconomic data and S&P 500 daily returns recover interpretable sparse coefficient maps and residual dependence graphs while automatically down-weighting crisis-period observations.

Sorted $\ell_1$ Penalized Estimation (SLOPE) models, that perform either variable or group selection, control the false discovery rate (FDR) under orthogonal settings with known noise, but such settings are rare in practice. Under general conditions, cross-validation is the default model selection approach for SLOPE, yet it targets predictive performance rather than FDR control. We address this gap for the SLOPE family of models by proposing new Bayesian approaches, Bayesian Group SLOPE (BGSLOPE) and Bayesian Sparse-group SLOPE (BSGS). BGSLOPE and BSGS embed group-based SLOPE models into a spike-and-slab framework, with BSGS providing a continuous spike-and-slab framework for sparse-group models. We further introduce Two-step Orthogonal (TSO), which transforms a general setting into an orthogonal one to recover SLOPE's FDR control properties. Through extensive synthetic and real data studies comparing all major model selection strategies for SLOPE models, the proposed Bayesian models consistently control FDR, achieve higher power, and outperform competing methods in prediction.

Fréchet regression provides a versatile framework for modeling responses in metric spaces with Euclidean predictors, yet current methodologies rely almost exclusively on frequentist approaches. We propose a Bayesian framework for Fréchet regression that offers a principled way of incorporating prior information into nonlinear global Fréchet regression. By targeting a novel Fréchet Bayes rule, we reduce the object-valued regression problem to a collection of tractable scalar regression tasks. Our approach allows for a controlled interpolation between the prior and the data-driven frequentist estimate, facilitating effective shrinkage toward informed values. While initially derived under Gaussian assumptions, we demonstrate that our framework is robust to model misspecification by establishing its validity under moment conditions via weak conditional expectations. The numerical properties of the proposed methodology are demonstrated in simulation studies and an application to microbiome compositional data, where we show that leveraging an auxiliary cohort to inform the prior significantly enhances predictive performance in a targeted, small-scale study

Electronic health records (EHR) pose large-scale multi-disease modeling problems in which many outcomes are rare and strongly influenced by shared risk factors. While modern approaches achieve strong predictive performance, they often treat diseases independently or rely on black-box architectures, offering limited insight into how risk factors organize disease risk and little principled uncertainty quantification. We introduce a Bayesian hypergraph inference framework that reframes multi-disease modeling around latent, risk-factor-modulated disease pathways. Risk factors act on hyperedges, latent disease subsets with shared risk patterns, allowing diseases to participate in multiple distinct pathways and enabling interpretable, higher-order structure beyond pairwise associations. A repulsion prior encourages parsimonious and identifiable structure, while posterior inference provides calibrated uncertainty over both disease groupings and risk-factor influence. To enable scalable inference on large EHR datasets, we develop a structured variational inference algorithm that preserves logical dependencies among hyperedge existence, disease membership, and pathway-level effects. Experiments on simulated data and UK Biobank demonstrate stable and interpretable disease pathway structure, well-calibrated uncertainty, improved estimation for rare diseases, and competitive predictive performance.

Bayesian optimization (BO) is a popular and effective approach for tuning expensive, noisy experiments, but requires the formulation of an explicit objective function. Preferential BO (PBO) removes this requirement by learning from pairwise human feedback, yet existing methods struggle to efficiently optimize beyond low- and medium-dimensional problems due to their global search approaches. We address this limitation by developing a family of local PBO methods that transfer key ideas from high-dimensional BO to the preferential setting. In particular, we introduce local PBO methods which adapt trust-region and derivative-informed local search to pairwise preference feedback, where the latter exploits first- and second-order derivatives of the Laplace-approximated GP posterior. Our benchmark on GP sample paths, standard optimization benchmark functions, and policy-search tasks shows that local PBO methods are especially effective in high-dimensional and complex landscapes with steep optima. Compared with global preference-based baselines, they can substantially reduce cumulative regret, making them particularly useful for real-world preference-based optimization tasks such as policy search.

While suitably scaled CNNs with Gaussian initialization are known to converge to Gaussian processes as the number of channels diverges, little is known beyond this Gaussian limit. We establish a large deviation principle (LDP) for convolutional neural networks in the infinite-channel regime. We consider a broad class of multidimensional CNN architectures characterized by general receptive fields encoded through a patch-extractor function satisfying mild structural assumptions. Our main result establishes a large deviation principle (LDP) for the sequence of conditional covariance matrices under Gaussian prior distribution on the weights. We further derive an LDP for the posterior distribution obtained by conditioning on a finite number of observations. In addition, we provide a streamlined proof of the concentration of the conditional covariances and of the Gaussian equivalence of the network. To the best of our knowledge, this is the first large deviation principle established for convolutional neural networks.

The Frequentist, Assisted by Bayes (FAB) framework constructs confidence regions that leverage prior information about parameter values. FAB confidence regions (FAB-CRs) have smaller volume for values of the parameter that are likely under the prior while maintaining exact frequentist coverage. This work introduces several methodological and theoretical contributions to the FAB framework. For Gaussian likelihoods, we show that the posterior mean of the mean parameter is contained in the FAB-CR. More generally, this result extends to the posterior mean of the natural parameter for likelihoods in the natural exponential family. These results provide a natural Bayes-assisted estimator to be reported alongside the FAB-CR. Furthermore, for Gaussian likelihoods, we show that power-law tail conditions on the marginal likelihood induce robust FAB-CRs that are uniformly bounded and revert to standard frequentist confidence intervals for extreme observations. We translate this result into practice by proposing a class of shrinkage priors for the FAB framework that satisfy this condition without sacrificing analytic tractability. The resulting FAB estimators equal prominent Bayesian shrinkage estimators, including the horseshoe estimator, thereby establishing insightful connections between robust FAB-CRs and Bayesian shrinkage methods.

Multivariate distributions that allow for asymmetry and heavy tails are important building blocks in many econometric and statistical models. The Unified Skew-t (UST) is a promising choice because it is both scalable and allows for a high level of flexibility in the asymmetry in the distribution. However, it suffers from parameter identification and computational hurdles that have to date inhibited its use for modeling data. In this paper we propose a new tractable variant of the unified skew-t (TrUST) distribution that addresses both challenges. Moreover, the copula of this distribution is shown to also be tractable, while allowing for greater heterogeneity in asymmetric dependence over variable pairs than the popular skew-t copula. We show how Bayesian posterior inference for both the distribution and its copula can be computed using an extended likelihood derived from a generative representation of the distribution. The efficacy of this Bayesian method, and the enhanced flexibility of both the TrUST distribution and its implicit copula, is first demonstrated using simulated data. Applications of the TrUST distribution to highly skewed regional Australian electricity prices, and the TrUST copula to intraday U.S. equity returns, demonstrate how our proposed distribution and its copula can provide substantial increases in accuracy over the popular skew-t and its copula in practice.

We consider a variant of the linear contextual stochastic multi-armed bandits, where the learner must provide recommendations to a group of users, each having its personalized preference vector, and in the presence of context distributions that are drifting over time. Under practitioner-friendly assumptions, we reduce this setting to linear bandit with stationary mean but heteroskedastic and non-stationary noise. We further study the case when the learner must ensure the mean reward of each decision must exceed that of a baseline strategy $\boldsymbolπ_0$ at each decision step. We introduce Dri-MED, an algorithm inspired from the linear version of the MED strategy, and carefully adapted to handle the non-stationary heteroskedastic noise. We show that the instance-dependent regret scales as $\tilde{\mathcal O}\left(\fracκ{\tildeΔ}d^2(\log(T)\right)$, where $\tildeΔ$ is the constraint-aware sub-optimality gap subject to policy $π_0$, with variance-aware multiplicative term $κ$ that we carefully handle using heteroskedastic regression. We further show Dri-MED enjoys $\tilde{\mathcal{O}}(d)$ expected constraint violations. Our numerical results suggest that Dri-MED significantly outperforms conservative baselines that ignores the drift and preference structure.

This paper proposes a synthetic control (SC) framework for the estimation of conditional distributional treatment effects. Identification rests on a parallel trends condition formulated in the parameter space of the semiparametric distribution regression (DR) model, which keeps the counterfactual conditional distribution within the model class. The weights solve a least-squares problem subject to an adding-up constraint, yielding a closed-form estimator. We derive the asymptotic distribution of the counterfactual estimator, with DR estimation error and weight estimation error contributing at the same rate to the asymptotic variance. Moreover, we propose a supremum test for the null of no treatment effect, whose limit is the supremum of a Gaussian process. Simulations illustrate that conditioning on covariates can reveal effects being difficult to detect from the unconditional distribution alone. An application to the 1992 New Jersey minimum wage increase using CPS data finds effects concentrated in the minimum-wage corridor for low-education, low-experience workers.

For approximately half of the individuals receiving mental health care, the results are suboptimal, even when treatments align with evidence-based guidelines. These limited effects may partly stem from how clinical decisions on treatment focus are made in mental health care. Typically, treatment strategy is guided by the diagnostic classification combined with the individualized case conceptualization. While standard, this approach may fall short for several reasons such as biases on the part of both the patient and therapist, and treatment guidelines being based on average effects that may not (exactly) suit the individual patient. To address these challenges, we propose a novel framework that reduces biases in clinical decision-making and makes it genuinely possible to tailor treatment focus to the individual patient. This framework involves (a) constructing causal graphs and estimating causal effects from intensively collected, longitudinal patient data, (b) simulating new time series based upon the causal relationships, and (c) using these simulations to identify the most effective treatment focus for the individual patient. By simulating and comparing different intervention strategies and examining both the estimated individual's responsiveness and its long-term effectiveness, this approach may generate useful insights to guide treatment focus and strategy, which can lead to a significant improvement of treatment outcomes in mental health care.

This paper proposes a collapsible method for estimating causal effects that maintains the estimator's consistency before and after marginalization over some variables in completed partially directed acyclic graphs (CPDAGs). We first introduce the estimate collapsibility for CPDAGs and characterize the minimal collapsible sets as strong d-convex hulls. An efficient algorithm is devised to obtain such sets in DAGs and is generalized to CPDAGs. Then, we combine the graph reduction procedure with the IDA framework. Finally, experiments and empirical analysis show the effectiveness of the collapsibility for causal estimations in CPDAGs. Code is available at https://github.com/Jamyang-D/strongly-convex.

Measuring causal effects in networked two-sided marketplaces is challenging due to treatment interference between market participants on different sides. When treatment is applied to one side (e.g., job seekers), their interactions with the other side (e.g., job posters) introduce spillover effects that violate the Stable Unit Treatment Value Assumption (SUTVA) and bias causal estimates. While cluster-based randomization mitigates this problem, prior approaches struggle with a fundamental trade-off: reducing spillover requires isolated clusters that will reduce the number of qualifying clusters, which decreases statistical power. This paper introduces EgoCluster V3, an iterative clustering algorithm that reduces spillover by 3x compared to prior versions while preserving node coverage and doubling test power. We further introduce MultiEgoCluster, which extends V3 through a two-stage procedure that first groups highly connected egos into multi-ego clusters before applying the iterative clustering algorithm. This achieves an additional ~56% spillover reduction and ~38% increase in sample size. Both methods are deployed in production at LinkedIn and have systematically enabled high-impact two-sided marketplace experiments. Since residual bias cannot be fully eliminated through clustering alone, we derive a theoretical bias correction method for average treatment effect (ATE) estimation based on graph structure and propose an approach to generalize results to the general population.

Generative AI and large language models can produce realistic predictions of human behavior from rich, unstructured inputs with little to no task-specific training data. Recent work uses these ``digital twin'' predictions to supplement human responses in surveys and experiments. We study the special case of using AI-generated predictions to reduce variance in randomized experiments. We argue that doing so requires no new estimators and that researchers can simply include AI predictions as covariates in standard regression adjustment, analogous to adjusting for a prognostic score. A benefit of this approach is a ``do no harm'' property whereby the adjusted estimator reverts to the unadjusted difference in means when predictions are uninformative. Other methods, such as variants of prediction-powered inference, do not have this guarantee. We provide implementation guidance, including how to obtain continuous scores from discrete LLM outputs and how to use LLMs to featurize unstructured inputs as auxiliary covariates. We demonstrate these ideas in simulations and three empirical applications: a survey mega-study, an email marketing A/B test, and a large-scale technology platform experiment. Overall, efficiency gains are real if modest, with greater benefits in studies that contain substantial text and other unstructured data. We also confirm the do no harm property empirically. Given these gains and limited costs, we recommend adjusting for AI-generated predictions as a regular empirical practice.

Externally controlled survival trials are increasingly used when concurrent randomized controls are infeasible, particularly in oncology and rare-disease settings with time-to-event endpoints. We target an average-treatment-effect-on-the-treated (ATT)-type marginal hazard-ratio estimand, comparing treatment with counterfactual control in the treated trial population, and estimate it using inverse-probability-weighted (IPW) Cox regression. Valid inference is challenging because IPW Cox regression depends on the weights through both event contributions and risk-set averages, making flexible machine-learning nuisance estimation difficult to incorporate directly. Building on machine-learning-assisted generalized entropy calibration (MEC) by Lee and Kim (2026), we propose MEC-Cox for ATT-weighted IPW Cox regression. The method begins with normalized source-propensity-score odds weights for external controls and then applies Bregman calibration to balance cross-fitted prognostic summaries between external controls and treated trial patients. The calibration basis may include control-survival predictions, Cox linear predictors, penalized-survival-model predictions, or other prognostic-score summaries. MEC-updated weights therefore play a dual role as source-transport and prognostic-score balancing weights. We establish consistency, characterize a calibration-induced efficiency gain, and develop a stacked sandwich variance estimator. Simulations show that MEC-Cox can reduce bias, increase efficiency, and improve coverage through flexible machine-learning-assisted adjustment.

We study causal discovery from observational data when some variables are hidden and the data-generating process follows a location-scale noise model (LSNM). Existing methods that handle hidden confounders typically assume additive noise, but in practice, causes often modulate not just the mean but also the variance of their effects. We prove that acyclic directed mixed graphs (ADMGs) satisfying a bow-free condition are identifiable under LSNM with hidden variables, establishing the first identifiability result for causally insufficient models beyond noise additivity. We further provide sufficient conditions for identifying causal direction even when the bow-free assumption is violated. Our two-stage algorithm, LSNM-UV, is sound and complete, and experiments demonstrate improved performance over additive baselines on heteroscedastic data.

Transfer learning addresses the challenge of transfering knowledge from one domain to another. Traditional transfer learning focuses on adapting models trained on a source domain (with a lot of observations) to improve performance on a target domain (with few observations). In this work we consider the case of a model shift and we focus on the transfer learning applied to a causal forest namely HTERF. This causal forest aims to estimate the Conditional Average Treatment Effect (CATE). The approach considered is the offset method presented by Wang (2016) adapted to a causal context. This method relies on the use of intermediate models in order to estimate the offset between source and target distributions. Our main result is a bound on the CATE error of HTERF on target depending on the error of the intermediate models. Simulation studies show the good performances of this approach in different settings on simulations and on a real-world dataset.

Longitudinal treatment decisions from multivariate time-series data require predicting potential outcomes under future treatment sequences in the presence of time-varying confounding, heterogeneous patient dynamics, and limited domain-specific data. Existing longitudinal causal estimators typically address this problem by training a new model for each cohort or simulator. We introduce Causal Longitudinal Prior-Fitted Networks (CausalLongPFN), a prior-fitted network for time-series causal inference in longitudinal treatment-response data and zero-shot in-context counterfactual outcome prediction. The model is pretrained entirely on synthetic episodes sampled from a broad prior over temporal structural causal models, exposing it to treatment-confounder feedback, latent heterogeneity, nonlinear state evolution, delayed effects, and cumulative treatment responses. At test time, CausalLongPFN remains frozen and is used zero-shot: it conditions on support trajectories, a query history, and a planned future treatment sequence, and returns a predictive distribution over future outcomes without gradient updates or propensity-model fitting. Multi-step predictions are obtained by recursively applying the one-step predictor under the specified treatment sequence. We evaluate the model on branchable cancer, HIV, and warfarin benchmarks with ground-truth counterfactual labels, and on factual-only rolling-origin prediction in MIMIC-III ICU trajectories. CausalLongPFN is competitive with domain-trained longitudinal baselines on counterfactual benchmarks and performs strongly on factual MIMIC-III prediction, suggesting that broad synthetic causal pretraining can provide a frozen, amortized alternative for zero-shot longitudinal treatment-response prediction when repeated domain-specific training is costly or impractical.

We consider continuous time to treatment initiation. This can commonly occur in preventive medicine, such as disease screening and vaccination; it can also occur with non-fatal health conditions such as HIV infection without the onset of AIDS. While traditional causal inference focused on `when to treat' and its effects, we consider the incremental causal effect when the intensity of time to treatment initiation is intervened upon. We derive the efficient influence function for this estimand and develop an estimation framework that accommodates flexible machine learning methods while achieving fast convergence rates. Valid confidence bands are obtained leveraging empirical process theory. We illustrate our approach via simulation, and apply it to cervical cancer screening data to study the incremental effect of time to subsequent HPV testing on cervical intraepithelial neoplasia detection.

Egocentric-Network Randomized Trials (ENRTs) are increasingly used to estimate causal effects under interference when measuring complete sociocentric network data is infeasible. ENRTs rely on egocentric network sampling, where a set of egos is first sampled, and each ego recruits a subset of its neighbors as alters. Treatments are then randomized across egos. While the observed ego-networks are disjoint by design, the underlying population network may contain edges connecting them, leading to contamination. Under a design-based framework, we show that the Horvitz-Thompson estimators of direct and indirect effects are biased whenever contamination is present. To address this, we derive bias-corrected estimators and propose a novel sensitivity analysis framework based on sensitivity parameters representing the probability or expected number of missing edges. This framework is implemented via both grid sensitivity analysis and probabilistic bias analysis, providing researchers with a flexible tool to assess the robustness of the causal estimators to contamination. We apply our methodology to the HIV Prevention Trials Network 037 study, finding that ignoring contamination may lead to underestimation of indirect effects and overestimation of direct effects.

Emerging single-cell technologies that combine CRISPR-based genetic perturbations with single-cell RNA sequencing, such as Perturb-seq, offer unprecedented opportunities to uncover cause-and-effect relationships among genes. Nonetheless, Perturb-seq experiments are subject to unobserved factors that, if not properly handled, can severely bias the inferred causal relationships between genes. These latent factors may arise not only from intrinsic molecular features of the regulatory elements, but also from unmeasured genes omitted due to cost-constrained experimental designs. Although methods for analyzing large-scale Perturb-seq data are rapidly maturing, approaches that explicitly account for such unobserved confounders when inferring causal gene networks are still lacking. Here, we propose a novel approach to accurately reconstruct causal gene networks from Perturb-seq data even when important confounders are missing. Our framework leverages proxy and instrumental variable strategies to exploit the rich information embedded in the perturbations, enabling unbiased estimation of the underlying directed acyclic graph (DAG) of gene expression. Applications to both comprehensive synthetic data and real CRISPR interference experiments in K562 cells demonstrate that our method outperforms baseline approaches that lack principled adjustments for unmeasured confounding, yielding more accurate and biologically relevant recovery of the true causal DAGs.

Modern treatment targeting methods often rely on estimating a conditional average treatment effect (CATE) using machine learning tools. While effective in identifying who benefits from treatment on the individual level, these approaches typically overlook system-level dynamics that may arise when treatments induce strain on shared capacity. We study the problem of targeting in Markovian systems, where treatment decisions must be made one at a time as units arrive, and early decisions can impact later outcomes through delayed or limited access to resources. We show that optimal policies in such settings compare CATE-like quantities to state-specific thresholds, where each threshold reflects the expected cumulative impact on the system of treating an additional individual in the given state. We propose an algorithm that augments standard CATE estimation with state-level value iteration to estimate these thresholds from observational data. Theoretical results establish consistency and convergence guarantees, and empirical studies demonstrate that our method improves long-run outcomes considerably relative to individual-level CATE targeting rules and generic offline reinforcement learning algorithms.

Randomized experiments are the gold standard for causal inference but face significant challenges in business applications, including limited traffic allocation, the need for heterogeneous treatment effect estimation, and the complexity of managing overlapping experiments. These factors lead to high variability in treatment effect estimates, making data-driven policy roll out difficult. To address these issues, we introduce the data pooling treatment roll-out (DPTR) framework, which enhances policy roll-out by pooling data across experiments rather than focusing narrowly on individual ones. DPTR can effectively accommodate both overlapping and non-overlapping traffic scenarios, regardless of linear or nonlinear model specifications. We demonstrate the framework's robustness through a three-pronged validation: (a) theoretical analysis shows that DPTR surpasses the traditional difference-in-mean and ordinary least squares methods under non-overlapping experiments, particularly when the number of experiments is large; (b) synthetic simulations confirm its adaptability in complex scenarios with overlapping traffic, rich covariates and nonlinear specifications; and (c) empirical applications to two experimental datasets from real world platforms, demonstrating its effectiveness in guiding customized policy roll-outs for subgroups within a single experiment, as well as in coordinating policy deployments across multiple experiments with overlapping scenarios. By reducing estimation variability to improve decision-making effectiveness, DPTR provides a scalable, practical solution for online platforms to better leverage their experimental data in today's increasingly complex business environments.

Standard approaches in generalizability often focus on generalizing the intent-to-treat (ITT). However, in practice, a more policy-relevant quantity is the generalized impact of an intervention across compliers. While instrumental variable (IV) methods are commonly used to estimate the complier average causal effect (CACE) within samples, standard approaches cannot be applied to a target population with a different distribution from the study sample. This paper makes several key contributions. First, we introduce a new set of identifying assumptions in the form of a population-level exclusion restriction that allows for identification of the target complier average causal effect (T-CACE) in both randomized experiments and observational studies. This allows researchers to identify the T-CACE without relying on standard principal ignorability assumptions. Second, we propose a class of inverse-weighted estimators for the T-CACE and derive their asymptotic properties. We provide extensions for settings in which researchers have access to auxiliary compliance information across the target population. Finally, we introduce a sensitivity analysis for researchers to evaluate the robustness of the estimators in the presence of unmeasured confounding and extend existing tests to evaluate instrument validity in this context. We illustrate our proposed method through extensive simulations and a study evaluating the impact of deep canvassing on reducing exclusionary attitudes.

Paired cluster-randomized experiments (pCRTs) are common in education program impact evaluation trials. Although common, there is surprisingly no clear consensus regarding how to analyze this randomization design to estimate average treatment effects. Variance estimation is also complicated due to the dependency created through pairing clusters. Therefore, we aim to provide an intuitive and practical comparison between different estimation strategies for pCRTs to inform practitioners' choice of strategy. To this end, we present a general framework for design-based estimation of an average individual effect in pCRTs. This framework offers a novel and intuitive view on the bias-variance trade-off between point estimators and emphasizes the benefits of covariate adjustment for estimation with pCRTs. In addition to providing a general framework for estimation with pCRTs, the point and variance estimators we present support fixed-sample unbiased estimation with similar precision to a common regression model and conservative variance estimation. Through simulation studies based on an educational efficacy trial, we compare the performance of the point and variance estimators reviewed. Our analysis and simulation studies inform the choice of point and variance estimators for analyzing pCRTs in practice.

This paper introduces a novel decomposition framework to explain heterogeneity in causal effects observed across different studies, considering both observational and randomized settings. We present a formal decomposition of between-study heterogeneity, identifying sources of variability in treatment effects across studies. The proposed methodology allows for robust estimation of causal parameters under various assumptions, addressing differences in pre-treatment covariate distributions, mediating variables, and the outcome mechanism. Our approach is validated through a simulation study and applied to data from the Moving to Opportunity (MTO) study, demonstrating its practical relevance. This work contributes to the broader understanding of causal inference in multi-study environments, with potential applications in evidence synthesis and policy-making.

We consider the problem of learning how to optimally allocate treatments whose cost is uncertain and can vary with pre-treatment covariates. This setting may arise in medicine if we need to prioritize access to a scarce resource that different patients would use for different amounts of time, or in marketing if we want to target discounts whose cost to the company depends on how much the discounts are used. Here, we show that the optimal treatment allocation rule under budget constraints is a thresholding rule based on priority scores (those with a higher score are treated first), and we propose a number of practical methods for learning these priority scores using data from a randomized trial. Our formal results leverage a statistical connection between our problem and that of learning heterogeneous treatment effects under endogeneity using an instrumental variable. We find our method to perform well in a number of empirical evaluations.

This paper studies the asymptotic behavior of sample canonical directions in a finite-rank spiked high-dimensional canonical correlation analysis model under a Gaussian population assumption. Under the asymptotic regime in which the dimensions of the two data blocks grow proportionally with the sample size, sample canonical directions are generally not consistent estimators of their population counterparts, even when the corresponding sample canonical correlations separate from the bulk spectrum. To quantify directional recovery, we investigate the squared alignment between a sample canonical direction and its associated population direction. For each simple population spike, we first establish a deterministic first-order limit for this squared alignment, which gives an explicit measure of the population-level directional information retained by the sample direction. We then prove a central limit theorem for its fluctuations around the deterministic limit, with an explicit asymptotic variance expressed through deterministic limits of resolvent trace functionals. To make the theoretical quantities computable from data, we further construct plug-in estimators for both the limiting mean and the asymptotic variance by inverting the deterministic outlier eigenvalue map, and prove their consistency. Numerical simulations and a real-data illustration support the theoretical results and demonstrate how the proposed estimators assess the recovery quality of sample canonical directions.

We study high-dimensional linear regression under a general symmetric convex constraint. Rather than imposing a specific sparsity-inducing penalty, we start from an arbitrary sign-symmetric and permutation-invariant convex body $K\subseteq \mathbb R^p$ and construct the sparse convexification hierarchy \[ K^{(s)} = \operatorname{conv}\{v\in K:\|v\|_0\le s\}. \] We propose a penalized least-squares estimator that searches over this hierarchy and adapts to the best sparse convex approximation of the target. Under standard sub-Gaussian assumptions on the random design and noise, we prove an oracle inequality showing that the estimator adapts to the best sparse convex approximation of the target. For an $s$-sparse target, the result yields a squared-error rate governed by the effective sparse dimension $s\log(ep/s)$, the noise level $σ$, and the Euclidean diameter $d_s$ of the sparse convexification $K^{(s)}$. The method applies broadly to symmetric norm balls and can be implemented using oracle access to the Minkowski functional of $K$. As a special case, the framework yields a consistency result for the constrained Lasso.

Undirected graphical models provide a fundamental framework for representing conditional independence structures among high-dimensional random variables. While undirected graphical model selection has become a central problem in high-dimensional statistics, most existing methods are restricted to parametric settings. In this paper, we develop a nonparametric approach to undirected graphical model selection based on diffusion models. Recent work has shown that diffusion models can adapt to the unknown graph structure of the underlying distribution, yet utilizing these models for explicit graph estimation remains unexplored. To bridge this gap, we introduce a novel diffusion-based method for nonparametric undirected graphical model selection. We establish the model selection consistency of the proposed method and demonstrate its empirical performance through extensive simulations and two real data analyses.

Gaussian graphical models in the spectral domain offer a principled approach for recovering conditional dependence structures in stationary high-dimensional time series. Inference on the spectral precision matrix at a fixed frequency enables tests of frequency-specific conditional associations among time series components. The problem is challenging because finite-sample discrete Fourier transforms induce truncation and smoothing biases, while the complex-valued nature of the spectral precision matrix complicates high-dimensional variance estimation, rendering methods for i.i.d. samples not directly applicable. Existing approaches do not provide full likelihood-based inference for the discrete Fourier transforms. We propose a high-dimensional inference framework for sparse spectral precision matrices using the full likelihood of neighboring discrete Fourier transforms. We construct a debiased complex graphical lasso estimator at any fixed frequency. Using asymptotic theory for quadratic forms of multivariate time series, we establish its asymptotic normality and construct entry-wise consistent covariance estimators by aggregating information across neighboring frequencies. The key theoretical contribution is the simultaneous control of regularization, finite-sample truncation, and smoothing biases, enabling valid inference. Simulation studies show reliable coverage away from zero frequency and improved detection power over the benchmark, with false discovery rates near the desired level.

We develop methodology for recovery of change points for data observed on more than one temporal index where changes may occur simultaneous in both indices, where the spatial component may be high dimensional. The work is motivated by climate monitoring problems where long series of data are available, e.g., daily observations (index 1) over several years (index 2). Such data may be evolving over the annual time scale, along with dynamic seasonal changes in the shorter time scale. We model this as a high dimensional mean process observed on a two dimensional grid with change points. Asymptotic estimation and inference results are developed under a single change point setup, including rates of convergence of the proposed method as well the resulting limiting distributions. The method is extended to the case of multiple changes. Theoretical results are supported numerically with monte-carlo simulations. We implement our work on a large scale climate data for the Pacific Northwest region of the United States.

Doubly-stochastic point processes model the occurrence of events over a spatial domain as an inhomogeneous Poisson process conditioned on the realization of a random intensity function. They are flexible tools for capturing spatial heterogeneity and correlation. However, existing implementations of doubly-stochastic spatial models are computationally demanding, often have limited theoretical guarantee, and/or rely on restrictive assumptions. We propose a penalized regression method for estimating covariate effects in doubly-stochastic point processes that is computationally efficient and does not require a parametric form or stationarity of the underlying intensity. Our approach is based on an approximate (discrete and deterministic) formulation of the true (continuous and stochastic) intensity function. We show that consistency and asymptotic normality of the covariate effect estimates can be achieved despite the model misspecification, and develop a covariance estimator that leads to a valid, albeit conservative, statistical inference procedure. A simulation study shows the validity of our approach under less restrictive assumptions on the data generating mechanism, and an application to Seattle crime data demonstrates better prediction accuracy compared with existing alternatives.

Probabilistic forecasters are increasingly learned, yet the baselines they are compared against are often weak or omitted. We show that the simplest possible conformal interval - a last-value point forecast wrapped in a finite-sample split-conformal residual quantile, with no parameters and no training - is a far stronger baseline than its near-total absence from recent learned-forecasting and conformal-time-series comparisons would suggest. In one-step-ahead online forecasting across 2,217 real series from nine public sources (Monash, LOTSA, the LTSF traffic/electricity/weather suites, METR-LA, BOOM, nips/probts), this ConformalNaive interval decisively beats the naive value-quantile baselines, the entire NPTS family (NPTS 73%, SeasonalNPTS 64% of series), and the published Conformal Seasonal Pools (CSP) method (71% of series, bootstrap 95% CI [69,73], paired Wilcoxon p approx 7.6e-135); it is on par with the simpler learned conformal predictors (RCI, quantile regression; median relative Winkler within 2%) and is beaten only by the adaptive-online and ensemble methods (SPCI, ACI, AgACI), which track distribution shift and lead by 9-33% relative Winkler. It is also better calibrated than a trained neural forecaster: on the six datasets that introduced DeepNPTS, the trivial floors cover the truth 84-85% of the time at a nominal 95%, versus DeepNPTS's 66%. At multi-step seasonal horizons the picture inverts: the random-walk floor is the weakest method and the seasonal pool (CSP) wins - a boundary we map. Finally we give ConformalNaive+, a one-line, training-free, horizon-adaptive selector that attains the better of two complementary floors at every horizon with restored coverage. We argue the matching conformal naive floor must be a mandatory baseline whenever a learned probabilistic forecaster claims gains.

Signed networks consist of both positive and negative relations, and structural balance theory provides an important conceptural framework for understanding their global tension structure. While existing statistical methods mainly focus on assessing empirical evidence of balance in a single observed network, many real-world signed relations evolve over time. This paper develops nonparametric inference for the population degree of structural balance at specified time points in dynamic signed networks, where the target time may or may not coincide with an observed snapshot. We consider a dynamic signed graphon model in which both edge formation and sign generation are governed by smoothly time-varying graphon functions. To exploit temporal smoothness, we construct a kernel-smoothed estimator that borrows information from snapshots near the target time point. Our theoretical analysis establishes a studentized inference procedure and a higher-order distributional approximation based on Edgeworth expansion, showing that temporal smoothing improves inference in sparse networks by reducing variance of observation noise, up to smoothing bias and time-discretization errors. We demonstrate the finite-sample performance and practical usefulness of the proposed method through extensive simulation studies and an application to a dynamic international relation network in political science.

We adopt the canonical polyadic (CP) decomposition to model high-dimensional tensor time series. Our primary goal is to identify and estimate the factor loadings in the CP decomposition. We propose a one-pass estimation procedure through standard eigen-analysis for a matrix constructed based on the serial dependence structure of the data. The asymptotic properties of the proposed estimator are established under a general setting as long as the factor loading vectors are linearly independent, allowing the factors to be correlated and the factor loading vectors to be not nearly orthogonal. The procedure adapts to the sparsity of the factor loading vectors, accommodates weak factors, and demonstrates strong performance across a wide range of scenarios. To further reduce estimation errors, we also introduce an iterative algorithm based on a novel double projection approach. We theoretically justify the improved convergence rate of the iterative estimator, and derive the associated limiting distribution. A consistent estimator of the asymptotic variance is also provided, which plays a key role in the related inference problems. All results are validated through extensive simulations and two real data applications.

This manuscript studies the problem of independence testing between two high-dimensional time series without assuming weak stationarity, that is, allowing their autocovariances to vary over time. To this end, we propose a bimodal weighted-average test statistic that removes the bias induced by temporal dependence under the null hypothesis, thereby avoiding the need to whiten the time series prior to hypothesis testing -- a procedure that is challenging in high-dimensional and nonstationary settings. To facilitate statistical inference, we develop a dependent wild bootstrap procedure. On the theoretical side, we derive a concentration inequality for quadratic forms of time series data stemming from a class of high-dimensional, nonlinear, and nonstationary processes. This result enables us to derive the asymptotic null distribution of the proposed test statistic and to establish the validity of the bootstrap algorithm. Numerical results show that the proposed test attains desired size and good power performance even when the dimension exceeds the sample size or when the data-generating process exhibits time-varying autocovariances. In contrast, tests based on whitening time series fail to maintain correct size in the presence of unstable autocovariance structures. Since nonstationary autocovariances commonly arise in real-life time series data, our work offers a robust procedure for independence testing.

Adaptive beamforming is a cornerstone of array signal processing, yet its performance often collapses in the face of complex, rapidly changing interference. When interferers appear or move unpredictably, conventional estimators encounter a fundamental memory trade-off: short windows enable rapid tracking but suffer from high estimation variance, while long windows provide stable rejection but fail to adapt to shifts. This challenge is resolved by introducing the Universal Switching Beamformer (USB), which integrates competitive sequential prediction into the beamforming architecture. By employing a linear transition diagram, the USB implicitly maintains an exponentially large family of candidate covariance histories and dynamically re-weights them based on their cumulative output power. This mechanism allows the beamformer to automatically vary its effective memory length without explicit change detection or heuristic parameter tuning. A theoretical upper bound is proven on the regret relative to an omniscient oracle that selects the best piecewise-stationary covariance model in hindsight. Extensive simulations and experiments on the SwellEx-96 dataset demonstrate that the USB achieves the agility of short-window estimators and the precision of long-term integration, providing a principled solution for tracking highly non-stationary scenes.

Accurately monitoring mental fatigue is critical for improving workplace safety and productivity. A recent study examined unobtrusively collected smartphone typing speed as a potential ambulatory proxy assessment of mental fatigue using data from the Intern Health Study (IHS). While population-level average typing speed patterns were found to be consistent with validated measures of mental fatigue, how these trajectories vary across participants and days may inform opportune moments for just-in-time interventions and remains an open question. Treating typing speed trajectories as sparsely observed functional data, we propose a novel sparse longitudinal functional principal component analysis (sparse LFPCA) method for decomposing variability and predicting individual curves. Specifically, sparse data are accommodated by casting covariance estimation as a structured penalized spline regression problem, enabling simultaneous estimation and smoothing of multiple covariance components while borrowing information across locations in the functional domain. Simulations show that sparse LFPCA (1) accurately estimates eigenfunctions and generates reasonable predictions for underlying curves, and (2) achieves similar or superior performance compared to existing alternatives. Our analysis of typing speed data collected from IHS reveals new and interpretable participant- and day-level patterns not captured by previous analyses and can be used to tailor behavioral interventions.

We study the asymptotic behaviour of widely used tests for evaluating and comparing predictive accuracy when forecast errors exhibit heavy tails. In particular, when loss differentials have infinite variance, the Diebold-Mariano test statistic converges to a nonstandard limit involving non-Gaussian stable random variables. As a consequence, conventional critical values can yield severely distorted inference: a nominal 5$\%$ test may reject a true null as often as 70$\%$ of the time. To establish these results, we develop a new stable limit theorem for strongly mixing, infinite-variance time series processes. Building on this theory, we consider sub-sampling-based inference that remains valid irrespective of tail-heaviness and requires no estimation of long-run variances or tail indices. An application to risk forecasts for emerging-market exchange rates shows that accounting for heavy tails can substantially alter conclusions about predictive performance relative to standard procedures.

We propose a simple estimator for the dynamic decomposition of the Generalized Dynamic Factor Model that avoids frequency-domain methods. First, we show that it is a reasonable approximation to assume that the dynamic common component of the Generalized Dynamic Factor Model admits a representation in terms of current and lagged statically pervasive factors. Then, assuming finite lag order, this simplification reduces estimation to a regression of the observed variables on estimated factors and their lags, where the factors are extracted via static principal components. The proposed approach naturally accommodates weak, non-pervasive factors within the dynamic common space. We establish consistency and asymptotic normality for both the dynamic and weak common components under a new asymptotic framework that allows for such weak factors. In an application to three high-dimensional time series panels of European macroeconomic data we detect a sizeable weak common component share in several key macroeconomic indicators.

We propose an information criterion for determining an unknown number of periodic components in functional time series. Identifying the number of frequencies in large-scale time series has been a central focus. To achieve this goal, we suggest an iterative procedure, utilizing the residual process obtained through least squares fitting. This iterative approach demonstrates broad applicability. We establish the consistency of the estimated number of periodic components by minimizing the information criterion. The efficacy of the procedure is illustrated through numerical simulations. In real data analysis, we apply this information criterion to temperature data and sunspot data.

The effectiveness of self-supervised learning (SSL) for physiological time series depends on the ability of a pretraining objective to preserve information about the underlying physiological state while filtering out unrelated noise. However, existing strategies are limited due to reliance on heuristic principles or poorly constrained generative tasks. To address this limitation, we propose a pretraining framework that exploits the information structure of a dynamical systems generative model across multiple time-series. This framework reveals our key insight that class identity can be efficiently captured by extracting information about the generative variables related to the system parameters shared across similar time series samples, while noise unique to individual samples should be discarded. Building on this insight, we propose PULSE, a cross-reconstruction-based pretraining objective for physiological time series datasets that explicitly extracts system information while discarding non-transferrable sample-specific ones. We establish theory that provides sufficient conditions for the system information to be recovered, and empirically validate it using a synthetic dynamical systems experiment. Furthermore, we apply our method to diverse real-world datasets, demonstrating that PULSE learns representations that can broadly distinguish semantic classes, increase label efficiency, and improve transfer learning.

We introduce a high-dimensional multiplier bootstrap for time series data based on capturing dependence through a sparsely estimated vector autoregressive model. We prove its consistency for inference on high-dimensional means under two different moment assumptions on the errors, namely sub-gaussian moments and a finite number of absolute moments. In establishing these results, we derive a Gaussian approximation for the maximum mean of a linear process, which may be of independent interest.

Multi-relational networks among entities are frequently observed in the era of big data. Quantifying the effects of multiple networks have attracted significant research interest recently. In this work, we model multiple network effects through an autoregressive framework for tensor-valued time series. To characterize the potential heterogeneity of the networks and handle the high dimensionality of the time series data simultaneously, we assume a separate group structure for entities in each network and estimate all group memberships in a data-driven fashion. Specifically, we propose a group tensor network autoregression (GTNAR) model, which assumes that within each network, entities in the same group share the same set of model parameters, and the parameters differ across networks. An iterative algorithm is developed to estimate the model parameters and the latent group memberships simultaneously. Theoretically, we show that the group-wise parameters and group memberships can be consistently estimated when the group numbers are correctly- or possibly over-specified. An information criterion for group number estimation of each network is also provided to consistently select the group numbers. Lastly, we implement the method on a Yelp dataset to illustrate the usefulness of the method.

We propose the data augmented bootstrap (DAB), a framework for constructing confidence intervals from approximately invariant transformations of the data. As special cases, DAB recovers popular methods that rely on exact group symmetries, such as conformal prediction, wild bootstrap for Maximum Mean Discrepancy U-statistics and the recently proposed SymmPI. Meanwhile, DAB also recovers the classical bootstrap method, which exploits the dataset's approximate invariance under uniform sampling of data indices as the dataset size grows. For all DAB methods, we establish theoretical coverage results that interpolate between finite-sample and asymptotic guarantees according to the strength of the invariance, and without assuming a group structure. The approximate invariance is measured in the Kolmogorov distance and, for statistics that satisfy Gaussian universality, reduces to conditional mean and variance matching. This allows us to incorporate data augmentation (DA), a widely used machine learning heuristic based on approximate invariances, into known statistical methods. We empirically test the performance of incorporating DA into bootstrap, wild bootstrap and conformal prediction for simulated settings as well as for image, language and scientific data.

Diffusion Monte Carlo (DMC) and Monte Carlo for particle transport with importance sampling both involve simulations of weighted walkers that undergo birth and death processes (splitting and Russian Roulette). The established implementations of these methods are quite different: Particle simulation Monte Carlo employs a stack to handle the splitting history whereas in traditional DMC one follows a swarm of walkers. The particle simulation Monte Carlo approach involves a depth first traversal of the visited configurations whereas the traditional DMC approach may be seen as a breadth first traversal. In the present work the implementation of a depth first, stack based approach to DMC is described and a complete code is presented. The depth first approach, called DMCD here, can be more memory efficient than the breadth first approach, both for total memory and for use of a memory hierarchy and of co-processors. The implementation appears very natural for population control and for descendant weighting and it unifies algorithmic treatment of the eigenvalue problem (DMC) with the linear equation problem (particle transport). A concern with DMCD that is not present in the breadth first approach, and that is successfully addressed here, is the need to maintain a pool of starters for use when a new walker is required and the stack is empty. The DMCD approach appears to have the potential to become the preferred implementation for many DMC applications.

Bayesian optimization (BO) is a widely used approach for black-box optimization that uses a Gaussian process (GP) as a surrogate and guides sequential evaluations via an acquisition function, with the ultimate goal of locating the global optimum $\mathbf{x}^{\star}$. To align with this goal, information-based acquisition functions such as Predictive Entropy Search (PES) model $\mathbf{x}^{\star}$ as a random variable and reduce the entropy of its distribution, but approximating this distribution via traditional GP posterior sampling is computationally expensive. To address this limitation, we leverage Conditional Diffusion Models (CDMs) to efficiently approximate the distribution of $\mathbf{x}^{\star}$ and develop BO-inherent training strategies for CDMs. Motivated by the structural properties of the CDM-learned distribution, we further develop an acquisition strategy termed Diffusion-based Mode Seeking (DMS) to guide the sequential evaluation. We establish a sub-optimality guarantee for the CDM-learned distribution and demonstrate through extensive experiments that DMS outperforms standard BO baselines.

Many model evaluation tasks reduce to estimating an average loss, error rate, or subgroup metric on a stratified pool when each label, human rating, or simulator call is costly. The precision-optimal Neyman allocation depends on within-stratum variances, which must be learned from the same observations used for estimation. We formulate this as a sequential allocation problem and use the exact one-step marginal variance reduction as the priority index. Replacing the unknown variances by independent inverse-chi-squared posterior draws yields TS-Neyman, a Thompson-sampling rule that preserves the oracle marginal-gain structure while randomizing over variance uncertainty. For any fixed finite number of strata, we prove almost-sure convergence of the TS-Neyman allocation proportions to the Neyman target, asymptotic optimality of the variance proxy, and a central limit theorem for the resulting adaptive stratified estimator. In two five-stratum budget-scaling benchmarks, one bounded-loss benchmark and one binary model-error benchmark in the spirit of Dai et al. 2023, TS-Neyman's relative efficiency stays within 5 percent of the oracle on the bounded-loss population and within about 15 percent on the binary benchmark. In an additional CivilComments real-data replay with confidence-based strata, it stays within about 8 percent of the oracle and improves on equal allocation by roughly 7 to 14 percent in MSE across budgets, while plug-in greedy and two-stage plug-in can degrade by over an order of magnitude under sparse pilots. Common-pilot warm-start and prior-sensitivity studies show that this behavior is stable under working-model and working-prior misspecification.

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.

While Bayesian inference has become increasingly popular with advances in computational resources, its algorithms can be computationally prohibitive and may not scale with large datasets. This has led to growing interest in alternative algorithms, such as approximation methods and variants of Markov chain Monte Carlo. Among these approaches, prior proposal-recursive Bayesian (PP-RB) inference facilitates scalable Bayesian computation by recursively updating the posterior distribution across stages and utilizing parallel computing resources. While the well-known ``degeneracy'' issue in PP-RB has been studied, another limitation that PP-RB can yield incorrect inferences when posterior distributions shift substantially between stages has remained unsolved. To address this, we propose parallel-tempered prior proposal-recursive Bayesian (PPP-RB) inference, which extends PP-RB by leveraging the key idea underlying Metropolis-coupled Markov chain Monte Carlo. We show both theoretically and empirically that PPP-RB targets the true posterior distribution. We illustrate PPP-RB through numerical studies and real data analysis in application to earthquake count data and sea surface salinity in the North Atlantic region. In these applications, we compare PPP-RB with PP-RB and a standard MCMC, demonstrating that PPP-RB is more efficient in terms of effective sample size per elapsed time.

Black-box variational inference (BBVI) is a methodology for posterior approximation that relies on stochastic optimization. In practice, the stochastic optimizers underpinning BBVI generally require extensive problem-specific tuning, which undermines its promise as a truly "black box" inference algorithm. However, over the past decade, many new adaptive stochastic optimization algorithms have been developed that reduce or remove entirely the need for tuning. In this work, we investigate this new collection of adaptive methods in the context of BBVI, with the goal of establishing the current state of the art in tuning-free optimization-based inference. In particular, we present a large-scale empirical evaluation of 56 stochastic gradient-based optimization algorithms applied to 1092 Bayesian inference optimization problems, involving over 550,000 individual optimization runs and 15 core-years of compute. The optimization algorithms we evaluate are chosen to represent a wide spectrum of recent approaches and the benchmark problems are chosen to span a range of difficulty, with posterior target dimension 1-10^4, condition number 1-10^8, and a range of variational families. Our results show that no single method dominates, but running a selection of 5 algorithms suffices to reliably get close to the best-possible observed performance. We thus provide a strong baseline for applications where expert tuning is not possible and for comparison when developing new stochastic optimization algorithms.

This paper extends MST-Direct, a Matching-via-Sinkhorn-Transport approach for multivariate geostatistical simulation, from the original bivariate, unconditional, small-grid formulation to multivariate, conditional, and large-grid settings. We address the three main limitations identified in the original work: (i) scalability beyond a few thousand nodes through a sparse, candidate-restricted Sinkhorn matcher with O(nC) memory complexity; (ii) extension to multiple variables by matching target value tuples onto an independent FFT-MA Gaussian backbone that reproduces a prescribed variogram; and (iii) hard-data conditioning by fixing observed data tuples at their spatial locations while conditioning the backbone through kriging. Because the transport plan remains a permutation of the target tuples, the multivariate joint distribution is preserved exactly. The method is validated using the same six-variate, heteroscedastic, strongly nonlinear reference distribution employed in Direct Multivariate Simulation (DMS), under both unconditional (200x200) and conditional (100x100, 200 hard-data samples) scenarios, and is benchmarked against the Projection Pursuit Multivariate Transform (PPMT). Results show that MST-Direct reproduces the joint distribution with zero histogram error, exactly honours hard data, and accurately reproduces the prescribed spatial correlation structure, whereas PPMT remains an approximation. Index Terms-Optimal transport, Sinkhorn algorithm, geostatistical simulation, multivariate simulation.

Manifold-constrained hyper-connections (mHCs) have recently been proposed as a principled extension of hyper-connections, where the residual mixing matrices are constrained to be doubly stochastic via projection onto the Birkhoff polytope. In practical mHC implementations, this constraint is enforced by Sinkhorn-Knopp iterations, and the backward pass relies on unrolling the iterative solver. This design introduces substantial computation and memory overhead, and may also yield inaccurate projections when the algorithm converges slowly on challenging inputs, undermining the intended norm-control and stability guarantees of mHCs. In this work, we focus on the practically important 4x4 Birkhoff projection setting and develop an end-to-end acceleration framework. By leveraging the dual formulation, we reduce the problem to a three-dimensional unconstrained convex problem and solve it with Newton's method, achieving fast convergence and high accuracy. For the backward pass, we replace the unrolled differentiation with implicit differentiation, yielding exact gradients without storing intermediate states. To exploit massive parallelism, we design a warp-level CUDA kernel that uses only register-level primitives, avoiding global and shared memory I/O. Extensive experiments against representative open-source baselines demonstrate that the proposed solver yields substantially more reliable doubly stochastic projections -- especially when the input magnitude is large -- and achieves significant end-to-end speedups (including the backward pass), reaching over 20x acceleration at large batch sizes while maintaining orders of magnitude smaller marginal errors.

When a statistical model $\{P_θ : θ\in Θ\}$ lacks analytically tractable likelihoods, parametric statistical inference based on data generated from an unknown underlying distribution $P$ can still be performed as long as simulations from the model are possible. This approach is called Simulation Based Inference (SBI). Statistical models are rarely exactly correct (that is, $P \notin \{P_θ: θ\in Θ\}$), and Robust SBI focuses on inferring a reasonable parameter even under model mis-specification. We focus on the setting where $P$ possesses potentially both geometric and Total Variation type discrepancies from $P_{θ^*}$. For this problem, we use a Kullback-Liebler informed robust Optimal Transport divergence, motivated by Empirical Likelihood considerations. We introduce a stochastic sub-gradient ascent algorithm with a convergence guarantee for estimating the semi-discrete version of this robust Optimal Transport divergence, and design a parallelized SBI algorithm which employs the regular bootstrap on top of minimum semi-discrete robust Optimal Transport for parameter uncertainty quantification. We demonstrate mathematically why the divergence is robust under a joint geometric plus Total Variation type contamination and then illustrate the robustness of inferences on a complex benchmark SBI task.

Support vector regression with Mean Absolute Percentage Error (MAPE) loss is theoretically well-motivated for forecasting applications where accuracy is evaluated in relative terms, but the sample-dependent dual box constraints it induces have not been addressed in the published SMO literature. We derive a Sequential Minimal Optimization algorithm for this setting and prove a structural-invariance result: the MAPE modification affects exactly two components of the SMO iteration -- working-set selection and analytic-update clipping -- leaving gradient bookkeeping and curvature computation identical to classical epsilon-SVR. Building on this invariance, we establish four efficiency improvements (asymmetric freeze-counters, warm-starting, block working-set updates of size four, and per-pair tolerance scaling) and resolve a previously-open convergence problem for the odd-symmetry kernel variant via adaptive spectral regularization. Numerical validation against three reference solvers across eleven synthetic configurations certifies solution agreement within standard tolerance. Wall-time benchmarks show the present algorithm achieves the lowest median runtime on every tested configuration against OSQP, MOSEK, and Clarabel. At production scale, the algorithm converges on the California Housing benchmark while the patched LIBSVM reference implementation reaches its iteration ceiling without satisfying optimality -- demonstrating the practical necessity of the theoretical efficiency mechanisms. An open-source R package and an explicit solver-adaptation recipe are provided.

Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning systematically discards geometrically misaligned directions to enhance this invariance proximity, which formally corresponds to the largest principal angle between the subspace and its image under the operator. Yet, existing techniques are largely restricted to Euclidean settings. To bridge this gap, this paper presents an approach for computing principal angles and vectors to enable Koopman subspace pruning within a Reproducing Kernel Hilbert Space (RKHS) geometry. We first outline an exact computational routine, which is subsequently scaled for large datasets using randomized Nystrom approximations. Based on these foundations, we introduce the Kernel-SPV and Approximate Kernel-SPV algorithms for targeted subspace refinement via principal vectors. Simulation results validate our approach.

Traditional filtering algorithms for state estimation -- such as classical Kalman filtering, unscented Kalman filtering, and particle filters -- show performance degradation when applied to nonlinear systems whose uncertainty follows arbitrary non-Gaussian, and potentially multi-modal distributions. This study reviews recent approaches to state estimation via nonlinear filtering based on conditional normalizing flows, where the conditional embedding is generated by standard MLP architectures, transformers or selective state-space models (like Mamba-SSM). In addition, we test the effectiveness of an optimal-transport-inspired kinetic loss term in mitigating overparameterization in flows consisting of a large collection of transformations. We investigate the performance of these approaches on applications relevant to autonomous driving and patient population dynamics, paying special attention to how they handle time inversion and chained predictions. Finally, we assess the performance of various conditioning strategies for an application to real-world COVID-19 joint SIR system forecasting and parameter estimation.

We present a consensus-based framework that unifies phase space exploration with posterior-residual-based adaptive sampling for surrogate construction in high-dimensional energy landscapes. Unlike standard approximation tasks where sampling points can be freely queried, physical systems with complex energy landscapes such as molecular dynamics (MD) do not have direct access to arbitrary sampling regions due to the physical constraints and energy barriers; the surrogate construction further relies on the dynamical exploration of phase space, posing a significant numerical challenge. We formulate the problem as a minimax optimization that jointly adapts both the surrogate approximation and residual-enhanced sampling. The construction of free energy surfaces (FESs) for high-dimensional collective variables (CVs) of MD systems is used as a motivating example to illustrate the essential idea. Specifically, the maximization step establishes a stochastic interacting particle system to impose adaptive sampling through both exploitation of a Laplace approximation of the max-residual region and exploration of uncharted phase space via temperature control. The minimization step updates the FES surrogate with the new sample set. Numerical results demonstrate the effectiveness of the present approach for biomolecular systems with up to 30 CVs. While we focus on the FES construction, the developed framework is general for efficient surrogate construction for complex systems with high-dimensional energy landscapes.

Finite-rank approximations are widely used to scale Gaussian process (GP) regression, but their posterior behavior can differ from that of the corresponding parent GP prior. We study a class of finite-rank GP priors built from locally supported basis expansions with dependent Gaussian coefficients. Our framework covers finite-element approximations based on the stochastic partial differential equation (SPDE) representation of Matérn GPs and regular-grid GP interpolation schemes. We show that, with a suitable prior on the resolution parameter $N$, these finite-rank expansions inherit the same posterior contraction rate as the corresponding parent GP prior under the same bandwidth specification used for that parent prior. Consequently, the interpolation construction under a squared-exponential parent GP attains the minimax-optimal rate up to logarithmic factors under a hierarchical prior on the bandwidth parameter and on $N$, while the SPDE construction attains the same rate under a bandwidth scaling depending on the sample size and the smoothness of the true function, together with a prior on $N$. We also develop a posterior sampler for the hierarchical interpolation model that jointly updates the resolution and bandwidth parameters, and we provide numerical studies that support the theory.

Stochastic localization is a pathwise analysis technique that has emerged as a powerful tool in high-dimensional probability and sampling. In this work, we extend stochastic localization to a joint framework for coupling probability measures and explore its applications in distributional data analysis. We first unify existing stochastic localization processes under Eldan's $α$-scheme and characterize their localization rates. Building on this, we introduce a joint scheme to couple probability measures via concurrent $α$-schemes driven by a shared Brownian motion. This construction is canonical and induces a family of metrics on the space of probability measures, which we call Eldan's $α$-distance. Alternative variants that extrapolate optimal Gaussian couplings to log-concave measures are also discussed. We study the theoretical properties of Eldan's $α$-distance, including its restriction to Gaussian measures and its behavior under affine transformations. For $α= 0$, we show it is topologically equivalent to the $2$-Wasserstein distance for measures supported on a common compact set; we also relate its weighted variants to linearized optimal transport in Wiener space and to score-matching objectives in training diffusion models. Computationally, we develop efficient estimators for Eldan's $α$-distance in the cases $α=0$ and $α=1/2$, with rigorous error guarantees for log-concave and finitely supported measures in the former setting and Gaussian measures in the latter. Finally, we apply Eldan's $α$-distance as a scalable surrogate for the $2$-Wasserstein distance to enable fast pairwise distance estimation and approximate computation of Wasserstein barycenters.

Federated Bayesian modeling requires combining evidence from distributed users into a coherent global posterior while keeping users' raw data on-device. We propose Federated Latent Graph MCMC (FLaG-MCMC), a computationally efficient framework for federated learning in which historical posterior samples of a shared global parameter are encoded into a learned low-dimensional latent space, connected via a $k$-nearest-neighbor graph, and transferred sequentially to new users as a nonparametric prior. Each user runs graph-based MCMC in the latent space guided by their own likelihood, returns updated global samples to the server, and retains local latent variables on-device. We demonstrate FLaG-MCMC on Bayesian meta-analysis for opioid use disorder prevalence estimation and on federated topic modeling, where the federated posterior closely approximates the pooled full-data posterior for both global parameters and local user-level inference.

We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem (UAT) for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

Feature interactions drive much of the predictive power of machine learning models, yet existing explanation methods only detect and quantify interactions without revealing their functional form, or visualize only restricted interaction types. We propose Surrogate-based Analysis of Interactions via Local effect Smooths (SAILS), a model-agnostic framework that analyzes pairwise interactions through interpretable generalized additive model (GAM) surrogates fitted to the local effects of a black-box model. For each interval of a feature of interest, the surrogate smooth terms isolate the interaction components on derivative level, enabling (i) interaction detection through a heuristic derived from significance tests on smooth terms, (ii) interaction form categorization into linear, product-separable, and non-product-separable types, and (iii) tailored, interpretable visualizations for each interaction type. We empirically validate the framework through controlled simulations and a real-world task, demonstrating its effectiveness for pairwise interactions, with limitations under strong feature correlations and higher-order interactions. SAILS fills a notable gap in the XAI toolbox, going beyond detection of interactions alone to characterizing their functional form.

We prove that $ρ\text{-}\mathrm{NPTS}_{\mathrm{SG}}$, an anchor-free nonparametric Thompson Sampling algorithm for risk-averse bandits, achieves regret matching the instance-dependent lower bound to leading order in $\log n$, establishing it as asymptotically optimal for any continuous risk functional $ρ$ (CVaR, mean-variance, Sharpe ratio, distortion risk measures, and more) on the class of distributions with bounded density and sub-Gaussian tails, including Gaussian arms. Both this result and its bounded-support counterpart require only continuity of $ρ$: strictly weaker than the dominance condition of prior parametric Thompson Sampling results, and strictly weaker than the Lipschitz condition of UCB-type algorithms, yielding the first instance-optimal guarantees for non-Lipschitz functionals such as the Sharpe ratio without parametric reward assumptions. The bounded-support case is developed first as a stepping stone sharing the same proof structure. The key technical contributions are a discretisation lemma (bounded support) and a truncated discretisation lemma (sub-Gaussian tails), each projecting the growing-alphabet Dirichlet posterior onto a fixed grid via the Dirichlet aggregation property, holding all polynomial prefactors at fixed degree independent of sample size and breaking the super-exponential barrier that blocked prior proofs.

Self-evolution offers a scalable path to stronger reasoning: a pretrained language model improves itself with only minimal external supervision. Yet existing methods either depend on extensively curated or teacher-generated training data, or, when the generator runs unsupervised, reward it by a difficulty heuristic that need not improve the solver. We introduce INFUSER, an iterative co-training framework with two co-evolving roles: a Generator that drafts questions and reference golden answers from a pool of unstructured, automatically collected documents, and a Solver that improves by training on them. The solver is trained with standard correctness rewards against the generator-provided answers, while the generator is rewarded by an optimizer-aware influence score that measures whether each proposed question would actually improve the solver on the target distribution. Because this continuous, noisy influence score is poorly served by standard GRPO, we propose DuGRPO, a dual-normalized variant of GRPO, for generator training. Together, these turn the document pool into an adaptive curriculum that favors questions useful to the current solver, not just hard ones. On Qwen3-8B-Base, INFUSER outperforms strong self-evolution baselines with over 20% relative improvement on Olympiad and SuperGPQA benchmarks, and an 8B INFUSER co-evolving generator outperforms a frozen 32B thinking generator on math and coding. Ablations confirm each design choice is necessary, and two extensions, applying INFUSER to an instruction-finetuned anchor and augmenting it with rule-verifiable RLVR data, further demonstrate the flexibility and generalizability of the framework. Code is available at https://github.com/FFishy-git/INFUSER.

Post-training quantization (PTQ) converts a trained full-precision model into low-bit weights without task-level retraining, while quantization-aware training (QAT) incorporates quantization into the training loop. Although PTQ is efficient and often accurate at moderate bitwidths, it can fail sharply at aggressive bitwidths; QAT is more expensive but can often recover the lost accuracy. We propose a unified geometric framework that explains both PTQ failure and QAT recovery. We model full-precision training as following a low-loss \emph{river} inside a wider \emph{valley}: a normal neighborhood of the river forms a nearly flat \emph{basin}, while leaving this basin incurs a sharp loss increase. When the quantization grid is comparable to the basin width, local PTQ objectives, including rounding and Hessian-based second-order reconstruction, can select a high-loss deployed quantized point outside the basin even when nearby low-loss quantized points exist. In this regime, straight-through-estimator-based QAT has a useful bias: it evaluates gradients at the deployed quantized weights while updating latent full-precision weights, causing the gradient to sense the valley wall and acquire an inward component that steers subsequent quantized iterates back into the basin. We formalize this mechanism through a local landscape model, construct a geometric PTQ failure mode, and prove finite-time QAT recovery under local quantizer-compatibility assumptions. Experiments across vision and language models under multiple neural-network quantization schemes corroborate the predicted basin-crossing failure of PTQ and the corresponding recovery mechanism of QAT.

We study a stochastic multi-armed bandit problem in which the set of available arms expands over time. This setting arises in sequential experimentation when new actions or treatments become available during an ongoing study, making regret against a single best arm in hindsight inappropriate. We instead evaluate performance relative to the best arm currently available, leading to a dynamic-regret criterion for arriving-arm environments. To address the resulting challenges of arrival information discrepancy (AID) and a drifting benchmark (DB), we propose UCB for Arriving Arms (UCB-AA), an elimination-based procedure with an aiding preliminary screening step for newly arrived arms before full competition with incumbent arms. We show that UCB-AA attains regret bounds that depend explicitly on the arrival process, achieves sublinear dynamic regret under regularity conditions on gap evolution, and admits an online extension for unknown horizons. Simulation results show that UCB-AA reduces wasted pulls and maintains a smaller active arm set while preserving competitive regret performance.

Recurrent neural networks maintain a hidden state $h_t$, but its probabilistic meaning is often unclear. We study hidden-state stability through \emph{backward coherence}: the extent to which $h_t$ can be reconstructed from $h_{t+1}$ by a learned backward projector $g_ϕ$. Under contraction and summable backward drift, the hidden-state sequence forms a quasi-reverse-martingale. This yields almost-sure convergence, rates under mixing, an interpretable limiting representation, finite pathwise stopping times, and a theoretical framework for time-uniform confidence sequences. Simulations support the theory. Backward-coherence regularisation reduces the empirical quasi-martingale total $\hat Q$ by $43$--$58%$, reaches stability $28$--$44%$ earlier than an unregularised RNN, and gives tracking-error recovery consistent with geometric bounds. Additional tests confirm echo-state forgetting rates bounded by $ρ$ and verify the increment-sum tube $R_t$ with $100%$ simultaneous coverage, although $R_t$ is conservative; in practice, the defect-tail proxy $\hat Q_t$ is the more useful monitor. The backward-coherence loss is also equivalent to minimising a Kullback--Leibler divergence in a Gaussian backward model, linking the method to variational inference. Extensions cover $ϕ$-mixing inputs, change-point tracking, and finite-sample concentration. Three real-data studies further validate the approach. On PhysioNet 2012 ICU data, the Reverse Martingale RNN (RMRNN) matches RNN mortality-prediction AUC while reaching stable representations 13 hours earlier. On FRED-MD, it reduces one-month-ahead forecast error by about fourfold under concept drift. On UCI Human Activity Recognition, it maintains lower post-transition tracking error with geometric decay. The guarantees apply under the stated assumptions; universality is not claimed.

Standard Reinforcement Learning with Verifiable Rewards (RLVR) training allocates a fixed rollout budget to every query, without regard for what each query's difficulty means for the current policy. This leads to two symmetric failure modes: easy queries produce near-zero advantage because the policy already solves them, while unsolvable queries produce no signal because the policy never solves them. Both regimes waste training FLOPs without contributing to a learning gradient. We introduce sorted Group Policy Optimization (sGPO), a compute-efficient strategy that trades a small budget of inference FLOPs for a large reduction in wasted training FLOPs. The key insight is that cheap inference compute can serve as a single offline proxy for query difficulty. By generating a small batch of parallel samples per query under the initial policy, we obtain a model-aware empirical success rate. This motivates setting the training rollout group size to the inverse of this success rate, a practical rule that maximizes sample efficiency by extracting the most advantage per generated rollout. This single profiling pass simultaneously drives data filtering (removing trivial queries and sub-sampling unsolvable ones), adaptive group size allocation, and curriculum construction (scheduling queries from easy to hard). sGPO matches or exceeds baseline performance while reducing total training compute by a factor of three, with the upfront inference profiling cost included.

Inference-Time Scaling (ITS) has largely succeeded in verifiable domains like math and coding, where cheap verification enables scalable output selection. However, extending ITS to tasks prone to systematic failure - driven by faulty initial assumptions or unmet multidimensional constraints - typically relies on costly external solvers or brittle, model-based verifiers. Our key insight is that the intrinsic statistics of parallel sample sets, specifically length-adjusted tail entropy, provide a robust discriminative signal for solution quality without access to ground truth. Crucially, these statistics serve as a difficulty gate for adaptive compute allocation, dynamically routing problems across scaling regimes. First, Intrinsic Selection (iS) ranks candidates post-hoc, matching consensus-based algorithms across three domains and improving engineering design selection by 20% over pass@1 baselines. Second, Intrinsic Particle Filtering (iPF) generalizes this to step-level resampling, guiding generation toward high-confidence reasoning trajectories to improve pass@1 by 6.1 points on average on hard math problems. Finally, Particle Distillation (dPF) injects privileged guidance via early logit blending and KL-guided resampling, steering generation past systematic reasoning errors to satisfy expert rubrics, yielding up to 26.5% gains on complex clinical responses. Our pipeline applies seamlessly across broad-purpose, domain-specialized, and multimodal architectures, successfully extending ITS to open-ended domains without requiring trained reward models or exact ground-truth verification.

Modern data-driven applications increasingly involve learning from multiple heterogeneous sources, where a target dataset is limited but related information is available across domains. Naively combining these sources can degrade performance when relevance varies or spurious signals are present, posing a fundamental challenge for trustworthy cross-domain learning. We propose Projection Transfer Learning (ProjectionTL), a unified framework that integrates hierarchical Bayesian modeling with adaptive projection for selective knowledge transfer. The key idea is to decouple transfer at two levels: first, we construct a source-guided hierarchical prior that aggregates information across sources using data-driven weights, capturing global alignment between each source and the target; second, we refine this borrowing through a posterior-projection step that operates at the feature level, selectively retaining coordinates that exhibit local agreement with the target signal. This two-stage design enables the method to simultaneously perform source selection and feature selection, thereby mitigating negative transfer while preserving interpretability. ProjectionTL provides a principled approach to integrating heterogeneous data across domains, bridging statistical modeling and modern machine learning paradigms for robust and interpretable transfer. Through simulations and real-world biomedical applications, we demonstrate improved accuracy, stability, and interpretability compared to existing methods. Our framework offers a scalable and generalizable strategy for trustworthy cross-domain learning in high-dimensional settings.

Pretrained models are often evaluated on multi-task leaderboards to measure their applicability in diverse contexts. However, current methods for aggregating performance across tasks into leaderboard-level rankings do not address the uncertainty and variability at the task level. While recent works have proposed interval-based model rankings, the principled aggregation of uncertainty from individual tasks to leaderboard-level rankings remains unaddressed, and variation in models' performance across tasks is frequently obscured. In this work, we introduce a hierarchical framework that constructs model rank intervals with statistical guarantees at both levels: task-level rank confidence intervals from pairwise comparisons, and leaderboard-level rank prediction intervals using a conformal approach. This enables reliable quantification of model rank for each observed task and for new potential tasks. Experiments on simulated data and the TabArena and PromptEval (MMLU) benchmarks show that our method yields statistically valid and informative intervals, enabling reliable, uncertainty-aware model ranking on leaderboards.

When a neural time-series model reports that one variable modulates another's effect on a target, is the discovered interaction a property of the data or an artifact of model flexibility? We argue that this is fundamentally a question of identifiability, governed by the geometry of the observed input support rather than by the specific neural architecture. We study the problem in a multiplicative-gating extension of neural additive vector autoregression (GNAVAR), in which source contributions are modulated by other lagged variables. We show that representational capacity is not identifiability: dependent inputs induce leakage between edge-specific interaction terms, and low-dimensional support permits distinct interaction decompositions that agree on the observed data while differing elsewhere. We then prove a population identifiability theorem for normalized minimal GNAVAR decompositions under explicit support conditions, including settings with shared modulators. The theory yields a simple practitioner-facing diagnostic: the effective rank of the joint lag-block covariance predicts, before fitting, whether interaction recovery is feasible for a given candidate set. When the candidate set is unknown, a two-seed stability check provides a practical operational test. The same support condition organizes empirical outcomes into the three states predicted by the theory. Our results show that interaction recoverability depends on support geometry, that effective rank provides a practical pre-fit diagnostic, and that instability across independent fits is a characteristic signature of non-identifiable interaction discovery. The identifiability phenomenon, the support condition, and the instability signature are model-agnostic; GNAVAR is the vehicle that makes them provable.

Muon replaces a matrix gradient $G=UΣV^\top$ by its polar factor $UV^\top$. This keeps the singular directions selected by the gradient, but makes the update spectrum flat. We study the optimization bias created by this operation. Under explicit alignment assumptions, we prove that the polar update is the one-step entropy-maximizing choice among bounded updates that use the gradient singular directions and do not adapt to the current weight spectrum. In an underdetermined regression model, we derive exact singular-value dynamics for continuous-time Muon and identify a measurement-dependent condition under which the normalized spectrum moves toward equal nonzero singular values. This geometry also rules out a common low-rank interpretation: at fixed Frobenius norm, Muon's distinguished state has a flat spectrum, whereas nuclear-norm minimization favors spectral concentration. Controlled matrix-sensing experiments separate the effect from simple gradient rescaling, show that norm-matched gradient descent does not reproduce Muon, and recover the predicted flattening trend across broad ablations. In small NanoGPT pretraining, Muon preserves stable rank, has a broad learning-rate plateau, and improves validation loss relative to AdamW; in a matched small-ViT control, the ranking reverses. The resulting picture is regime-dependent: Muon is not universally superior, but its flat-spectrum bias can help when many spectral directions need to remain active.

Compositional priors describe the generic properties of layered functions in deep Bayesian models, where deep neural networks with random weights are a canonical example.In the wide-network limit, the prior is a Gaussian process with a depth-dependent kernel, and its behaviour as depth grows has been extensively studied through this kernel. Here, we study another case, where each layer itself is a vector valued Gaussian process, and our aim is similarly to understand the limiting behaviour of the prior as depth grows. Previous GP work has established that for the RBF kernel and a certain range of bandwidths $r$, the prior degenerates in the limit, converging to the set of constant functions -- which is not useful as a probabilistic model. In this paper we establish several new results. First, we identify a sharp bandwidth threshold $r_c(d) = Θ(\sqrt{d})$ above which the limit is degenerate, strengthening the earlier bounds. Second, and more importantly, we show that for $r$ below the threshold $r_c(d)$ the prior converges to a limit distribution $π_{\bar{Z}}$. We also prove that these distributions are non-degenerate and non-Gaussian, with non-vanishing dependence between coordinates. In contrast to the previously known degenerate regime, deep Gaussian process priors can therefore admit non-trivial limits. Empirically, we verify the threshold across a range of dimensions $d$, and demonstrate a complex multimodal behaviour of the limit distributions $π_{\bar{Z}}$ -- a regime that becomes increasingly narrow with $d$ and would be hard to identify without knowing the threshold.

Reinforcement Learning from Human Feedback via Proximal Policy Optimization often suffers from policy mode collapse, brittle exploration loops, and distribution drift. This paper introduces Variational Proximal Policy Optimization (\(\textsc{VP}_2\textsc{O}\)), a particle-based variational inference framework that maps policy optimization to Stein Variational Gradient Descent within a Mixture-of-Experts architecture. By leveraging functional kernels over localized expert prototypes alongside an expert orthogonalization loss, \(\textsc{VP}_2\textsc{O}\) introduces a geometry-based proximal-control mechanism that can reduce reliance on fixed clipping or KL schedules. Our results on a 33B/4B sparse Mixture-of-Experts model show several improvements across complex reasoning benchmarks, establishing a \(+\mathbf{179}\) ELO gain on Codeforces and a \(\mathbf{32\%}\) reduction in token count on AIME mathematical reasoning tasks.

Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps. In Euclidean space, barycentric projection converts a coupling into a map by taking conditional expectations, but on a Riemannian manifold curvature and cut loci make this operation nontrivial. We develop a framework for barycentric projections of transport couplings on Riemannian manifolds. The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss. The corresponding minimum value is an integrated conditional Fréchet variance, which vanishes exactly for map-induced couplings and therefore defines a conditional-variance Monge defect. We also study a tangential log-exp projection, prove its Euclidean exactness, its compatibility with Brenier-McCann maps in the Monge case, and its interpretation as the first unit Riemannian gradient update for the intrinsic objective. For discrete couplings, both constructions decompose row-wise into weighted Fréchet mean and log-exp problems. Experiments on spherical data, synthetic SPD data, and real EEG covariance matrices support the proposed division of roles: the intrinsic projection is the variational representative, while the tangential projection is a useful local displacement surrogate.

Performative prediction studies feedback loops that arise when predictive models are deployed in consequential domains. In these settings, deploying a model can change the population whose patterns the model aims to predict, inducing a distribution shift that is endogenous to the learning system. This perspective departs from classical treatments of distribution shift, where shifts are typically modeled as exogenous changes in the data-generating process. Yet, in practice, distribution shift is rarely one or the other. Predictive models may influence future data through the decisions they support, while the world itself continues to drift for reasons beyond the learner's control. We study partially performative prediction, a framework that captures both endogenous and exogenous sources of distribution shift. The framework generalizes performative prediction by allowing the data distribution to evolve both in response to the deployed model and according to an external, time-varying process. We extend the central notions of performative stability and performative optimality to this setting by defining their online analogues that track the evolving partially performative environment. We analyze practical learning heuristics, including repeated retraining, and characterize when they successfully adapt to partially performative environments.

Accurate vessel traffic flow prediction is crucial for smart port operations and navigational safety. However, maritime traffic flow data are often highly sparse with intermittent bursts, making robust forecasting challenging. Under such conditions, conventional spatio-temporal graph neural networks (ST-GNNs) can degrade toward conservative near-zero predictions and fail to capture non-zero activity. Although zero-inflated negative binomial (ZINB) models partially address excess zeros, their two-part formulation can still remain conservative around abrupt transitions. To address these issues, we propose a model-agnostic learnable Tweedie head that can be attached as a plug-and-play output module to arbitrary ST-GNN backbones. Instead of likelihood-based Tweedie training, which typically requires surrogate objectives, our approach optimizes the closed-form Tweedie unit deviance and predicts the mean for point forecasting while learning a node-level variance power to capture heterogeneous variability across port areas. Experiments on a maritime traffic graph constructed from real-world AIS data in the Port of Los Angeles and Long Beach show that the proposed head consistently improves RMSE across multiple ST-GNN backbones, especially on non-zero events, leading to more reliable forecasts for practical maritime traffic control.

Real-world datasets across image and text domains are often characterized by skewed class distributions and noisy annotations, which jointly degrade model performance, particularly on minority classes. Among existing solutions, active learning offers an effective and efficient paradigm by selectively querying the most informative and balanced samples for annotation. We propose an innovative active learning framework that mitigates class imbalance and selects the most informative samples to annotate. Leveraging foundation model priors, our algorithm enables imbalance-aware co-decisions between foundation model and small model to tackle noisy and imbalanced labels across various domains. We introduce the first study to systematically explore active learning under the dual challenges of label noise and class imbalance across image and text domains. Extensive experiments on imbalanced datasets demonstrate that our method achieves substantial annotation savings-over 50% compared to the best active learning baseline-while preserving performance and robustness to label noise.

We investigate how to make small tabular foundation models effective for High-Dimensional, Low-Sample Size (HDLSS) tabular prediction without retraining large backbones. We introduce Graph-guided Ordering with Local Refinement (GO-LR), show its equivalence to weighted Minimum Linear Arrangement, and interpret the practical solver as a TSP-path-style surrogate. We propose GOTabPFN,which builds on GO-LR, and a Neuro-Inspired Subunit Compression (NSC) unit to pool locally adjacent ordered features into meta-features, yielding a compact representation that makes TabPFN-style prediction practical in HDLSS regimes. Across tabular benchmarks, GOTabPFN improves stability and accuracy under tight token budgets.

Agentic reinforcement learning (RL) enables LLM agents to improve continuously from environment rewards, yet the resulting policies do not systematically accumulate reusable strategies that generalize across tasks. Modular skills can provide such reusable strategies, yet existing skill-augmented RL methods decouple skill creation from policy optimization, risking adopting skills that conflict with the evolving policy. Inspired by Anthropic's Skill Creator, we introduce ReSkill, an RL-in-the-loop skill creation framework that reconciles skill evolution with policy learning. ReSkill exploits the group-wise structure of GRPO to naturally embed three mechanisms with only marginal additional overhead: (1) an assertion-driven skill creator that diagnoses failures from past experience and proposes conditional, trigger-based skill revisions; (2) within-group rollout sampling that enables controlled comparison of skill versions, capturing which version best supports the policy's ongoing learning; and (3) Thompson Sampling with adaptive discounting to balance exploration and exploitation in skill version selection as the policy evolves. Across several domains, ReSkill consistently outperforms existing memory and skill-based RL methods, with the largest gains on unseen tasks. Analysis of the skill lifecycle shows skills being automatically created, tested, refined, and pruned as the policy improves, demonstrating reconciled skill-policy co-evolution.

We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NODEs (SA-NODEs). We constructively prove that SA-NODEs are capable of \emph{exact} interpolation of admissible finite datasets, and even satisfy a stronger property that we call \emph{simultaneous cell controllability} (SCC): their flows can map prescribed disjoint cells into arbitrarily small target balls. This property is the mechanism that upgrades interpolation into quantitative generalization, by allowing SA-NODEs to emulate piecewise-constant nonparametric estimators. Consequently, our risk bounds recover the rates of histogram and nearest-neighbor estimators, provided the network width satisfies a conservative scaling with the sample size. Numerical experiments show that trained SA-NODEs achieve competitive -- often lower -- test errors than these baselines. Finally, we show that the explicit time dependence is essential. Although two-layer autonomous NODEs can interpolate geometrically nondegenerate datasets, structural obstructions prevent them from achieving SCC. These limitations, further confirmed numerically, support the view that SA-NODEs provide a minimal effective architecture for learning.

Uncertainty quantification (UQ) is critical for the deployment of machine learning predictors in real-world scenarios where the data distribution may shift over time (i.e., data may not be exchangeable). Online conformal prediction (OCP) methods address this issue at the expense of either (i) group-wise error control or (ii) learning-rate independent implementation. Group-conditional coverage is essential for fairness across different collections of data points and for providing finer UQ guarantees. Parameter-free optimization is crucial for robustness to adversarial and unknown data shifts. We propose a parameter-free algorithm for group-conditional OCP and demonstrate that it achieves the best group-conditional coverage guarantees. We evaluate our algorithm on synthetic and real-world data, demonstrating that our method not only improves the reliability of existing parameter-free OCP methods but also provides prediction intervals that are comparable in size to well-tuned group-conditional approaches. By unifying group-conditional coverage with parameter-free online algorithms, our work lays a foundation for fair and robust uncertainty quantification in shifting environments.

The classic concept of "calibrated forecasts" and its more recent refinement, "calibeating," are defined with respect to the standard quadratic scoring rule. We extend these notions to the class of $\textit{proper}$ scoring rules (for which the best forecast is the true distribution) and define $\textit{proper-calibration}$ and $\textit{proper-calibeating}$ by requiring the errors to converge to zero uniformly over all bounded proper scoring rules. We first establish that calibration always implies proper-calibration, whereas calibeating need not imply proper-calibeating. Second, we show how to guarantee proper-calibeating and proper-multicalibeating. Finally, we demonstrate the equivalence between proper-calibration and universal no regret when best replying to forecasts in decision-making under uncertainty.

Deep learning models are widely deployed in safety-critical domains, but remain vulnerable to adversarial attacks. In this paper, we study the adversarial robustness of NTK neural networks in the context of nonparametric regression. We establish minimax optimal rates for adversarial regression in Sobolev spaces and then show that NTK neural networks, trained via gradient flow with early stopping, can achieve this optimal rate. However, in the overfitting regime, we prove that the minimum norm interpolant is vulnerable to adversarial perturbations.

Recent vision backbones, such as Transformer families and state-space models like Mamba, have achieved remarkable progress on image recognition. Despite their empirical success, these architectures remain far from the computational principles of the human brain, often demanding enormous amounts of training data while offering limited interpretability. We propose the Vision Hopfield Memory Network (V-HMN), a brain-inspired vision backbone that integrates hierarchical memory mechanisms across layers with iterative refinement updates. Specifically, V-HMN incorporates local Hopfield modules that provide associative memory dynamics at the image patch level, global Hopfield modules that function as episodic memory for contextual modulation, and a predictive-coding-inspired refinement rule for iterative error correction. By organizing these memory-based modules hierarchically, V-HMN captures both local and global dynamics in a unified framework. Memory retrieval exposes the relationship between inputs and stored patterns, providing a prototype-based form of interpretability through explicit memory retrieval, while the reuse of stored patterns improves data efficiency. This brain-inspired design therefore enhances data efficiency and provides a prototype-based form of interpretability compared to existing self-attention- or state-space-based approaches. We conducted extensive experiments on public image classification benchmarks. V-HMN achieves strong performance on small- and medium-scale benchmarks, and remains competitive with widely adopted backbone architectures on ImageNet despite minimal architectural tuning, while offering improved data efficiency and a prototype-based form of interpretability. These findings highlight the potential of V-HMN as a memory-centric alternative to standard vision backbones, thereby bridging brain-inspired computation with modern machine learning.

When predictors are statistically dependent, the appropriate definition of feature importance depends on the operational goal. Conditional-incremental measures are well-suited for feature selection, acquisition, and compression, where shared predictive information is treated as redundancy. For post-hoc interpretation, however, the goal is often to attribute predictive signals across correlated measurement channels. We introduce Disentangled Feature Importance (DFI), a population-level attribution framework for this setting. DFI maps covariates to an independent latent representation under a specified entropic optimal transport geometry, computes latent importance, and attributes it back to the original covariates through barycentric sensitivities. We show that broad conditional-incremental FI functionals target conditional incremental predictive value under squared-error loss, and therefore answer a different question from attribution of shared predictive signal under dependence. Under fixed transport cost, reference law, and regularization level, DFI defines a well-specified family of estimands. Latent scores admit a functional ANOVA interpretation, and in the Gaussian linear case, the attributed DFI recovers the classical $R^2$ decomposition for correlated regressors. We derive influence-function-based inference under nuisance-rate and smoothness conditions, and show in simulations and an HIV-1 neutralization-resistance analysis that DFI yields stable, interpretable, uncertainty-quantified attributions of shared predictive signal.

Gaussian process (GP) bandits provide a powerful framework for performing blackbox optimization of unknown functions. The characteristics of the unknown function depend heavily on the assumed GP prior. Most work in the literature assume that this prior is known but in practice this seldom holds. Instead, practitioners often rely on maximum likelihood estimation to select the hyperparameters of the prior - which lacks theoretical guarantees. In this work, we study two algorithms for joint prior selection and regret minimization in GP bandits based on GP Thompson sampling (GP-TS): Prior-Elimination GP-TS (PE-GP-TS) that disqualifies priors with poor predictive performance, and HyperPrior GP-TS (HP-GP-TS) that utilizes a bi-level Thompson sampling scheme. We theoretically analyze the algorithms and establish a sublinear regret bound for HP-GP-TS. In addition, we demonstrate the effectiveness of these algorithms compared to the alternatives through extensive experiments with synthetic and real-world data.

Machine learning model performance improvements tend to arise from competition and application. For deployment, we consider prescriptive scaling laws: given a pre-training compute budget, what downstream accuracy is attainable with contemporary post-training practice, and how stable is that mapping as the field evolves? Using large-scale observational evaluations with 5k existing and 2k newly evaluated model checkpoints spanning 2022-2026 across six benchmarks, we estimate capability boundaries, high conditional quantiles of benchmark scores as a function of log pre-training FLOPs, via smoothed quantile regression with a monotone, saturating sigmoid parameterization. We validate temporal reliability by fitting on earlier model generations and evaluating on later releases: across four of six tasks, the out-of-distribution coverage error remains below 2%, while math reasoning exhibits a consistently advancing boundary over time. For instance, at a budget of 10^24 FLOPs, the estimated attainable accuracies are 0.83 on IFEval and 0.54 on MATH Lvl 5. We then extend our approach to analyze task-dependent saturation and to probe contamination-related shifts on math reasoning tasks. Finally, we introduce a balanced I-optimal sampling algorithm that recovers near-full-data frontiers using roughly 20% of the parameter-count-weighted evaluation budget, as low as 5% on some tasks, while maintaining comparable calibration. Together, our work releases Proteus-2k, the latest model performance evaluation dataset, and introduces a practical methodology for translating compute budgets into reliable performance expectations and for monitoring when capability boundaries shift across time.

We study offline reinforcement learning under $Q^\star$-approximation and partial coverage, a setting that motivates practical algorithms such as Conservative $Q$-Learning (CQL; Kumar et al., 2020) but has received limited theoretical attention. Our work is inspired by the following open question: "Are $Q^\star$-realizability and Bellman completeness sufficient for sample-efficient offline RL under partial coverage?" We answer in the negative via an information-theoretic lower bound. To identify additional structure that enables sample-efficient offline RL under partial coverage, we introduce a general decision-estimation framework, inspired by model-free decision-estimation coefficients (DEC) for online RL (Foster et al., 2023b; Liu et al., 2025b). Our framework decomposes offline RL complexity into decision complexity and value estimation error. This allows modular study of both sub-problems. Our result not only unifies existing results (Chen and Jiang, 2022; Uehara et al., 2023), but further improves and generalizes them. On the decision complexity side, our improvement includes: the first $ε^{-2}$ sample complexity bound for soft $Q$-learning under partial coverage that improves Uehara et al.'s (2023) $ε^{-4}$ bound, the removal of the need for additional online interaction in the value-gap setting of Chen and Jiang (2022), and new learnable settings beyond the above two cases. On the value estimation side, we provide a new characterization of the role of Bellman completeness under partial coverage, and the first characterization of offline learnability for general low-Bellman-rank MDPs (Jiang et al., 2017; Du et al., 2021; Jin et al., 2021). The latter is a canonical online RL setting that has remained unexplored in offline RL except for special cases. As a side contribution, our techniques give the first analysis of CQL in the function approximation setting.

Performative predictions influence the very outcomes they aim to forecast. We study performative predictions that affect a sample (e.g., only existing users of an app) and/or the whole population (e.g., all potential app users). This raises the question of how well models generalize under performativity. For example, how well can we draw insights about new app users based on existing users when both of them react to the app's predictions? We address this question by embedding performative predictions into statistical learning theory. We prove generalization bounds under performative effects on the sample, on the population, and on both. A key intuition behind our proofs is that in the worst case, the population negates predictions, while the sample deceptively fulfills them. We cast such self-negating and self-fulfilling predictions as min-max and min-min risk functionals in Wasserstein space, respectively. Our analysis reveals a fundamental trade-off between performatively changing the world and learning from it: the more a model affects data, the less it can learn from it. Moreover, our analysis results in a surprising insight on how to improve generalization guarantees by retraining on performatively distorted samples. We illustrate our bounds in a case study on prediction-informed assignments of unemployed German residents to job trainings, drawing upon administrative labor market records from 1975 to 2017 in Germany.

We introduce Wedge Sampling, a new non-adaptive sampling scheme for low-rank tensor completion. We study recovery of an order-$k$ low-rank tensor of dimension $n \times \cdots \times n$ from a subset of its entries. Unlike the standard uniform entry model (i.e., i.i.d. samples from $[n]^k$), wedge sampling allocates observations to structured length-two patterns (wedges) in an associated bipartite sampling graph. By directly promoting these length-two connections, the sampling design strengthens the spectral signal that underlies efficient initialization, in regimes where uniform sampling is too sparse to generate enough informative correlations. Our main result shows that this change in sampling paradigm enables polynomial-time algorithms to achieve both weak and exact recovery with nearly linear sample complexity in $n$. The approach is also plug-and-play: wedge-sampling-based spectral initialization can be combined with existing refinement procedures (e.g., spectral or gradient-based methods) using only an additional $\tilde{O}(n)$ uniformly sampled entries, substantially improving over the $\tilde{O}(n^{k/2})$ sample complexity typically required under uniform entry sampling for efficient methods. Overall, our results suggest that the statistical-to-computational gap highlighted in Barak and Moitra (2022) is, to a large extent, a consequence of the uniform entry sampling model for tensor completion, and that alternative non-adaptive measurement designs that guarantee a strong initialization can overcome this barrier.

We analyze the Accelerated Noisy Power Method, an algorithm for Principal Component Analysis in the setting where only inexact matrix-vector products are available, which can arise for instance in decentralized PCA. While previous works have established that acceleration can improve convergence rates compared to the standard Noisy Power Method, these guarantees require overly restrictive upper bounds on the magnitude of the perturbations, limiting their practical applicability. We provide an improved analysis of this algorithm, which preserves the accelerated convergence rate under much milder conditions on the perturbations. We show that our new analysis is worst-case optimal, in the sense that the convergence rate cannot be improved, and that the noise conditions we derive cannot be relaxed without sacrificing convergence guarantees. We demonstrate the practical relevance of our results by deriving an accelerated algorithm for decentralized PCA, which has similar communication costs to non-accelerated methods. To our knowledge, this is the first decentralized algorithm for PCA with provably accelerated convergence.

It is folklore that reusing training data more than once can improve the statistical efficiency of gradient-based learning. While this phenomenon has been extensively studied in linear regression, the benefit of multi-pass gradient descent (GD, which reuses all the data) over one-pass stochastic gradient descent (online SGD, which uses each data point only once) is not well-understood in nonlinear and non-convex settings, except for a loss modification mechanism achieved by the first two passes on the data. In this work, we consider learning a $d$-dimensional single-index model with a quadratic activation, for which it is known that one-pass SGD requires $n\gtrsim d\log d$ samples to achieve weak recovery. We first show that this $\log d$ factor in the sample complexity persists for full-batch spherical GD on the correlation loss; however, by simply truncating the activation, full-batch GD exhibits a favorable optimization landscape at $n \simeq d$ samples, thereby outperforming one-pass SGD (with the same activation) in statistical efficiency. We complement this result with a trajectory analysis of full-batch GD on the squared loss from small initialization, showing that $n \gtrsim d$ samples and $T \gtrsim\log d$ gradient steps suffice to achieve strong (exact) recovery.

Flow-based methods have achieved significant success in various generative modeling tasks, capturing nuanced details within complex data distributions. However, few existing works have exploited this unique capability to resolve fine-grained structural details beyond generation tasks. This paper presents a flow-inspired framework for representation learning. First, we demonstrate that a rectified flow trained using independent coupling is zero everywhere at $t=0.5$ if and only if the source and target distributions are identical. We term this property the \emph{zero-flow criterion}. Second, we show that this criterion can certify conditional independence, thereby extracting \emph{sufficient information} from the data. Third, we translate this criterion into a tractable, simulation-free loss function that enables learning amortized Markov blankets in graphical models and latent representations in self-supervised learning tasks. Experiments on both simulated and real-world datasets demonstrate the effectiveness of our approach. The code reproducing our experiments can be found at: https://github.com/probabilityFLOW/zfe.

The performance of large language models (LLMs) on verifiable tasks is usually measured by pass@k, the probability of answering a question correctly at least once in k trials. At a fixed budget, a more suitable metric is coverage@cost, the average number of unique questions answered as a function of the total number of attempts. We connect the two metrics and show that the empirically-observed power-law behavior in pass@k leads to a sublinear growth of the coverage@cost (diminishing returns). To solve this problem, we propose Reset-and-Discard (ReD), a query method of LLMs that increases coverage@cost for a given budget, regardless of the pass@k form. Moreover, given a pass@k, we can quantitatively predict the savings in the total number of attempts using ReD. If pass@k is not available for the model, ReD can infer its power-law exponent. Experiments on three LLMs across coding (HumanEval), math (GSM8K), and reasoning (MMLU-Pro) benchmarks demonstrate that ReD substantially reduces the required attempts, tokens, and USD cost to reach a desired coverage, while also offering an efficient way to measure inference power-laws. ReD's advantage is maintained for imperfect verifiers and outperforms the tested allocation baselines.

Operator models are regression algorithms between Banach spaces of functions. They have become an increasingly critical tool for spatiotemporal forecasting and physics emulation, especially in high-stakes scenarios where robust, calibrated uncertainty quantification is required. We introduce Local Sliced Conformal Inference (LSCI), a distribution-free framework for generating function-valued, locally adaptive prediction sets for operator models. We prove finite-sample validity and derive a data-dependent upper bound on the coverage gap under local exchangeability. On synthetic Gaussian-process tasks and real applications (air quality monitoring, energy demand forecasting, and weather prediction), LSCI yields tighter sets with stronger adaptivity compared to conformal baselines. We also empirically demonstrate robustness against biased predictions and certain out-of-distribution noise regimes.

Network pruning is used to reduce inference latency and power consumption in large neural networks. However, most methods focus on empirical results at the expense of understanding the pruning process. We introduce Hyperflux, a novel $L_0$ method which models pruning as a continuously evolving system determined by flux, the gradient response to a weight's removal, and pressure, a global regularization driving weights toward pruning. By exploiting this model, Hyperflux's pruning behavior becomes understandable at both microscopic (weight regrowth/pruning) and macroscopic (sparsity convergence, etc.) levels. We also introduce a novel pressure scheduler that reliably targets desired sparsities. Hyperflux achieves competitive results with ResNet-50, VGG-19 and DeiT-T/S on CIFAR-10, CIFAR-100 and ImageNet datasets.

We develop a unified statistical framework for softmax-gated Gaussian mixture of experts (SGMoE) that addresses three long-standing obstacles in parameter estimation and model selection: (i) non-identifiability of gating parameters up to common translations, (ii) intrinsic gate-expert interactions that induce coupled differential relations in the likelihood, and (iii) the tight numerator-denominator coupling in the softmax-induced conditional density. Our approach introduces Voronoi-type loss functions aligned with the gate-partition geometry and establishes finite-sample convergence rates for the maximum likelihood estimator (MLE). In over-specified models, we reveal a link between the MLE's convergence rate and the solvability of an associated system of polynomial equations characterizing near-nonidentifiable directions. For model selection, we adapt dendrograms of mixing measures to SGMoE, yielding a consistent, sweep-free selector of the number of experts that attains pointwise-optimal parameter rates under overfitting while avoiding multi-size training. Simulations on synthetic data corroborate the theory, accurately recovering the expert count and achieving the predicted rates for parameter estimation while closely approximating the regression function. Under model misspecification (e.g., $ε$-contamination), the dendrogram selection criterion is robust, recovering the true number of mixture components, while the Akaike information criterion, the Bayesian information criterion, and the integrated completed likelihood tend to overselect as sample size grows. On a maize proteomics dataset of drought-responsive traits, our dendrogram-guided SGMoE selects two experts, exposes a clear mixing-measure hierarchy, stabilizes the likelihood early, and yields interpretable genotype-phenotype maps, outperforming standard criteria without multi-size training.

Oversampling is one of the most widely used approaches for addressing imbalanced classification. The core idea is to generate additional minority samples to rebalance the dataset. Most existing methods, such as SMOTE, require converting categorical variables into numerical vectors, which often leads to information loss. Recently, large language model (LLM)-based methods have been introduced to overcome this limitation. However, current LLM-based approaches typically generate minority samples with limited diversity, reducing robustness and generalizability in downstream classification tasks. To address this gap, we propose a novel LLM-based oversampling method designed to enhance diversity. First, we introduce a sampling strategy that conditions synthetic sample generation on both minority labels and features. Second, we develop a new permutation strategy for fine-tuning pre-trained LLMs. Third, we fine-tune the LLM not only on minority samples but also on interpolated samples to further enrich variability. Extensive experiments on 10 tabular datasets demonstrate that our method significantly outperforms eight SOTA baselines. The generated synthetic samples are both realistic and diverse. Moreover, we provide theoretical analysis through an entropy-based perspective, proving that our method encourages diversity in the generated samples.

We study the fundamental problem of calibrating a linear binary classifier of the form $σ(\hat{w}^\top x)$, where the feature vector $x$ is Gaussian, $σ$ is a link function, and $\hat{w}$ is an estimator of the true linear weight $w^\star$. By interpolating with a noninformative $\textit{chance classifier}$, we construct a well-calibrated predictor whose interpolation weight depends on the angle $\angle(\hat{w}, w_\star)$ between the estimator $\hat{w}$ and the true linear weight $w_\star$. We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle $\angle(\hat{w}, w_\star)$ can be consistently estimated. Furthermore, the resulting predictor is uniquely $\textit{Bregman-optimal}$, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.

Self-supervised learning (SSL) learns representations from massive unlabeled data, yet the resulting models typically operate as black boxes, necessitating domain-specific explanations. We introduce KREPES, a unified framework to analytically interpret the learned representations of SSL objectives, including SimCLR, BYOL, and VICReg. By bridging empirical neural tangent kernel approximations of neural networks with the Representer Theorem for kernels, we express the learned latent space directly via "Representer Landmarks", which are the representations of influential unlabeled training examples. We introduce novel metrics, "Sample-Specific Influence Score", "Concept-Conditioned Influence Score" and "Feature Alignment Gap", to quantify the transparency of the learned representations. KREPES enables direct audit of the latent space without supervision, for example, revealing an algorithmic bias in the Adult-1M dataset where SSL uses demographic proxies for income. Finally, to ensure scalability to benchmarks with 1M+ samples (ImageNet-1K, Adult-1M), KREPES introduces a novel Nyström approximation-based analytical inference framework for SSL objectives.

We investigate the finite-time convergence properties of Temporal Difference (TD) learning with linear function approximation, a cornerstone of reinforcement learning. We are interested in the so-called ``robust'' setting, where the convergence guarantee does not depend on the potential function's minimal curvature. While prior work has established convergence guarantees in this setting, these results typically rely on the artificial assumption that each iterate is projected onto a bounded set. Removing such a condition was left as an open problem by Bhandari et al. (COLT'18), hypothesizing the need for additional ``regularity conditions''. In this paper, we show that the simple unprojected TD(0) converges with a rate of $\widetilde{\mathcal{O}}\left(\frac{\|θ^*\|^2_2}{\sqrt{T}}\right)$ in expectation, even in the presence of Markovian noise. We do not require an additional regularity condition, but only a minor polylog correction to the learning rate. Our analysis reveals a novel self-bounding property of the TD updates and exploits it to guarantee bounded iterates.

The Gromov-Wasserstein (GW) variant of optimal transport, designed to compare probability densities defined over distinct metric spaces, has emerged as an important tool for the analysis of data with complex structure, such as ensembles of point clouds or networks. To overcome certain limitations, such as the restriction to comparisons of measures of equal mass and sensitivity to outliers, several unbalanced or partial transport relaxations of the GW distance have been introduced in the recent literature. This paper is concerned with the Conic Gromov-Wasserstein (CGW) distance introduced by Séjourné, Vialard, and Peyré. We provide a novel formulation in terms of semi-couplings, and extend the framework beyond the metric measure space setting, to compare more general network and hypernetwork structures. With this new formulation, we establish several fundamental properties of the CGW metric, including its scaling behavior under dilation, variational convergence in the limit of volume growth constraints, and comparison bounds with established optimal transport metrics. We further derive quantitative bounds that characterize the robustness of the CGW metric to perturbations in the underlying measures. The hypernetwork formulation of CGW admits a simple and provably convergent block coordinate ascent algorithm for its estimation, and we demonstrate the computational tractability and scalability of our approach through experiments on synthetic and real-world high-dimensional and structured datasets.

This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.

In this paper, we focus on the matching recovery problem between a pair of correlated Gaussian Wigner matrices with a latent vertex correspondence. We are particularly interested in a robust version of this problem such that our observation is a perturbed input $(A+E,B+F)$ where $(A,B)$ is a pair of correlated Gaussian Wigner matrices and $E,F$ are adversarially chosen matrices supported on an unknown $εn * εn$ principal minor of $A,B$, respectively. We propose an approximate message passing (AMP) type iterative algorithm that succeeds in polynomial time as long as the correlation $ρ$ between $(A,B)$ is a non-vanishing constant and $ε= o\big( \tfrac{1}{(\log n)^{20}} \big)$. A key distinction from standard AMP is the introduction of a time-dependent matrix multiplication step within the iteration, which simultaneously enlarges the feature dimension and cancels the correlation during the iteration. The main methodological inputs for our result are the iterative random graph matching algorithm proposed in \cite{DL22+, DL23+} and the spectral preprocessing procedure proposed in \cite{IS24+}. To the best of our knowledge, our algorithm is the first efficient random graph matching type algorithm that is robust under any adversarial perturbations of $n^{1-o(1)}$ size.

A central question in vector- and function-valued learning is how to design kernels that capture both local and non-local interactions while remaining computationally tractable. Existing operator-valued kernels offer only partial answers: separable kernels are efficient but fail to model interactions across the function domain, while commutative kernels capture only pointwise structure. To address this, we propose spectral truncation kernels, a new class of positive definite kernels for vector- and function-valued learning based on spectral truncation and $C^*$-algebra. By allowing noncommutative products in the kernel construction, the proposed kernels induce interactions across the data function domain and fill the gap between existing separable and commutative kernels. In addition, by using the $C^*$-algebraic framework, we reduce the computational cost compared to the existing vector-valued RKHS framework with operator-valued kernels.

It is becoming increasingly common in regression to train neural networks that model the entire distribution even if only the mean is required for prediction. This additional modeling often comes with performance gain and the reasons behind the improvement are not fully known. This paper investigates a recent approach to regression, the Histogram Loss, which involves learning the conditional distribution of the target variable by minimizing the cross-entropy between a target distribution and a flexible histogram prediction. We design theoretical and empirical analyses to determine why and when this performance gain appears, and how different components of the loss contribute to it. Our results suggest that the benefits of learning distributions in this setup come from improvements in optimization rather than modelling extra information. We then demonstrate the viability of the Histogram Loss in common deep learning applications without a need for costly hyperparameter tuning.

Embedding high-dimensional data into a low-dimensional space is an indispensable component of data analysis. In numerous applications, it is necessary to align and jointly embed multiple datasets from different studies or experimental conditions. Such datasets may share underlying structures of interest but exhibit individual distortions, resulting in misaligned embeddings using traditional techniques. In this work, we propose Entropic Optimal Transport (EOT) eigenmaps, a principled approach for aligning and jointly embedding a pair of datasets with theoretical guarantees. Our approach leverages the leading singular vectors of the EOT plan matrix between two datasets to extract their shared underlying structure and align them in a common embedding space. We interpret our approach as an inter-data variant of the classical Laplacian eigenmaps and diffusion maps embeddings, showing that it enjoys many favorable analogous properties. We analyze a generative model in which two observed high-dimensional datasets share latent variables supported on a common low-dimensional manifold, while each dataset is subject to translation, geometric distortion, orthogonal nuisance structure, and noise. In a large-sample, high-dimensional regime, we prove that the EOT plan concentrates around a population kernel on an effective manifold determined by the geometric mean of the distortions, with invariance to translations, orthogonal nuisance structure, and noise. Subsequently, we relate our embedding to eigenfunctions of population-level operators encoding the density and geometry of the shared manifold. Finally, we showcase the performance of our approach for data integration and embedding through simulations and analyses of real-world biological data, demonstrating its advantages over alternative methods in challenging scenarios.

Large-scale black-box models have become ubiquitous across numerous applications. Understanding the influence of individual training data sources on predictions made by these models is crucial for improving their trustworthiness. Current influence estimation techniques involve computing gradients for every training point or repeated training on different subsets. These approaches face obvious computational challenges when scaled up to large datasets and models. In this paper, we introduce and explore the Mirrored Influence Hypothesis, highlighting a reciprocal nature of influence between training and test data. Specifically, it suggests that evaluating the influence of training data on test predictions can be reformulated as an equivalent, yet inverse problem: assessing how the predictions for training samples would be altered if the model were trained on specific test samples. Through both empirical and theoretical validations, we demonstrate the wide applicability of our hypothesis. Inspired by this, we introduce a new method for estimating the influence of training data, which requires calculating gradients for specific test samples, paired with a forward pass for each training point. This approach can capitalize on the common asymmetry in scenarios where the number of test samples under concurrent examination is much smaller than the scale of the training dataset, thus gaining a significant improvement in efficiency compared to existing approaches. We demonstrate the applicability of our method across a range of scenarios, including data attribution in diffusion models, data leakage detection, analysis of memorization, mislabeled data detection, and tracing behavior in language models. Our code will be made available at https://github.com/ruoxi-jia-group/Forward-INF.

Flow Matching has become a cornerstone of modern generative models like Stable Diffusion 3, largely due to the efficiency of its Rectified Flow (RF) variant. The success of RF hinges on iteratively learning straight trajectories, pushing generation towards fewer sampling steps. However, the theoretical link between path geometry and sampling efficiency has been underexplored. This paper fills this gap by introducing a novel \textit{Piecewise Straightness} parameter, $γ_{2,T}$. We establish the first Wasserstein convergence bound that explicitly links the discretization error of \textit{any} general flow-model to $γ_{2,T}$, proving that minimizing curvature is the key to achieving high-fidelity, one-step sampling. Building on this theory, we establish the first theoretical framework to analyze the straightness of RF. We begin by offering intuitive geometric arguments for simple cases before identifying sufficient conditions under which a single rectification step (1-RF) yields a perfectly straight or even a Monge optimal coupling. While whether these sufficient conditions are met depends on the problem geometry, they enable the first concrete proofs in this area. Critically, fulfilling these conditions makes the subsequent flow (2-RF) perfectly straight ($γ_{2,T}=0$). This eliminates the discretization error in our bound and makes flawless, single-step sampling possible.

We present a manifold-based machine learning encoder-decoder method for learning dynamics in time, notably partial differential equations (PDEs), in which the manifold latent space evolves according to Ricci flow. This can be accomplished by parameterizing the latent manifold stage and subsequently simulating Ricci flow in a physics-informed setting, matching manifold quantities so that Ricci flow is empirically achieved. We emphasize dynamics that admit low-dimensional representations. With our method, the manifold, induced by the metric, is discerned through the training procedure, while the latent evolution due to Ricci flow provides an accommodating representation. By use of this flow, we sustain a canonical manifold latent representation for all values in the ambient PDE time interval continuum. We showcase that the Ricci flow facilitates qualities such as learning for out-of-distribution data and adversarial robustness on select PDE data. Moreover, we provide a thorough expansion of our methods in regard to special cases which allow higher-dimensional representations, such as Ricci flow on the hypersphere and neural discovery of non-parametric geometric flows with entropic strategies.

Thompson sampling has proven effective across a wide range of stationary bandit environments. However, as we demonstrate in this paper, it can perform poorly when applied to non-stationary environments. We show that such failures are attributed to the fact that, when exploring, the algorithm does not differentiate actions based on how quickly the information acquired loses its usefulness due to nonstationarity. Building upon this insight, we propose predictive sampling, an algorithm that deprioritizes acquiring information that quickly loses usefulness. Theoretical guarantee on the performance of predictive sampling is established through a Bayesian regret bound. We provide versions of predictive sampling for which computations tractably scale to complex bandit environments of practical interest. Through numerical simulations, we demonstrate that predictive sampling outperforms Thompson sampling in all non-stationary environments examined.

Ultra-high-dimensional array-based CpG methylation studies require statistical frameworks that simultaneously provide supervised structure discovery, interpretability, scalable latent-dimension identification, and computational feasibility. We propose SOLAR (Supervised Orthogonal Low-rank Adaptive Regression), a supervised low-rank latent-factor framework for identifying CpG-level methylation structure associated with residualized DNAm age. SOLAR combines orthogonal low-rank regression with a penalized maximum a posteriori formulation, dimension-adaptive BIC-type penalization, and a trans-dimensional simulated-annealing strategy for automatic latent-rank selection, together with theoretical guarantees including identifiability, fixed-rank recovery, and rank-selection consistency under suitable regularity conditions. The framework additionally incorporates computationally and memory-efficient optimization strategies demonstrating scalability up to $p=10^7$, while analyses at $p=10^6$ remain feasible on standard desktop computing environments. Simulation studies demonstrate stable rank recovery, competitive supervised signal recovery, and strong scalability across moderate-, high-, and ultra-high-dimensional regimes. Using longitudinal EPIC-array CpG methylation data from the GUSTO birth cohort, comprising $n=1051$ methylation profiles collected across infancy and early childhood with approximately 860,000 assayed CpGs per sample, SOLAR identifies heterogeneous supervised methylation structure associated with residualized DNAm age beyond chronological age alone, together with biologically coherent CpG signatures and enrichment patterns.

Class imbalance is common when developing clinical prediction models (CPMs) and is often assumed to lead to poor predictive performance. Several methods have been proposed to correct data imbalance during CPM development. However, it remains unclear whether correcting class imbalance improves or harms CPM performance. This study investigated how imbalance correction affects classification performance and prediction stability. We simulated the development and internal validation of CPMs using penalised logistic regression under different imbalance-correction strategies, including algorithm-level rebalancing, data-level rebalancing by oversampling, and combined over- and under-sampling. The simulation dataset was derived from the GUSTO-I trial, which included 40,830 patients and 2,851 events. All imbalance-correction strategies were evaluated across sample-size scenarios ranging from 500 to 40,830. Model performance and prediction stability were assessed using 200 bootstrap resamples, including discrimination, calibration, calibration stability, mean absolute prediction error (MAPE), and classification instability index (CII). Class imbalance correction did not meaningfully improve model discrimination. Both data-level and algorithm-level correction led to miscalibration, risk overestimation, and increased prediction instability, as shown by prediction stability, MAPE, and CII plots, compared with models developed without correction. These findings suggest that class imbalance correction does not necessarily improve CPM performance and may compromise calibration and prediction stability. Class imbalance should not be treated as a pathology that automatically requires correction. In clinical prediction modelling, routine imbalance correction by default is generally not advisable.

Heterogeneous susceptibility models for epidemic dynamics preferentially assume that individual susceptibility follows a gamma distribution, which permits analytical reduction to a low-dimensional system. However, the true empirical distributional form in any given population is unknown. Here we investigate the consequences of misspecifying the susceptibility distribution by comparing gamma and lognormal specifications in a Susceptible-Exposed-Infectious-Removed (SEIR) framework. When both distributions are matched on mean and coefficient of variation ($ν$), we find that their epidemic trajectories diverge once heterogeneity is moderate or high ($ν\gtrsim 1$), with the lognormal producing a later, larger peak and a greater final size. We then assess the impact of distributional misspecification on statistical inference. Using synthetic datasets, we fit correctly specified and misspecified models by maximum likelihood. In a default scenario, where inference is based on simulated data for a single epidemic, both models can reproduce the data by compensating through correlated shifts in heterogeneity and intervention parameters. When inference is based on two simulated epidemics, however, this compensation may be reduced by known constraints of how parameters are related across epidemics. In these cases, the correctly specified model recovers all parameters accurately, while the misspecified model tends to give biased estimates. These inference biases propagate into forecasts, but predictions remain relatively accurate when compared to homogeneous models which more than double peak incidences in scenarios where $ν\approx 1$, for instance. We conclude that deviations resulting from the susceptibility distribution misspecifications assessed here are minor and encourage the adoption of heterogeneous models in future epidemic forecasting.

Externally controlled trials (ECTs), including single-arm studies augmented with historical data and hybrid randomized designs with partial external augmentation, are increasingly used when concurrent randomized controls are infeasible or unethical. Regulatory guidance from the FDA, EMA, and NMPA calls for sensitivity analysis of borrowing assumptions, yet provides no structured template for which analyses to run or how to interpret them together. We propose a three-pillar framework organized around three questions: was the borrowing appropriate, did it contribute meaningful value, and are the conclusions robust to perturbation? The framework comprises eight modular analyses covering heterogeneity diagnostics, source influence, no-borrowing references, effective sample size, prior sensitivity, tipping points, alternative borrowing methods, and structural model sensitivity. It is method-agnostic and applies to both Bayesian and frequentist borrowing in patient-level or hybrid settings. We illustrate the framework using simulated data that mimic a hybrid evidence synthesis from a historical approval of ethnic-bridging submission under a real-world-evidence regulatory pathway. That original analysis combined individual patient data from a global pivotal study and a regional real-world study with aggregate data from two published cohorts, fitted via a Bayesian longitudinal model with ethnic-difference parameters. The worked example provides a reproducible template for sensitivity analysis in ECT submissions.

Understanding behavioural responses to disturbances is vital for wildlife conservation. For example, in the Arctic, the decrease in sea ice has opened new shipping routes, increasing the need for impact assessments that quantify the distance at which marine mammals react to vessel presence. This information can then guide targeted mitigation policies, such as vessel slow-down regulations and delineation of avoidance areas. Using telemetry data to determine distances linked to deviations from normal behaviour requires advanced statistical models, such as threshold hidden Markov models (THMMs). While these are powerful tools, they do not assess whether the estimated threshold reflects a meaningful behavioural shift. We introduce a lasso-penalized THMM that builds on computationally efficient methods to impose penalties on HMMs and present a new, efficient penalized quasi-restricted maximum-likelihood estimator. Our framework is capable of estimating thresholds and assessing whether the disturbance effects are distinguishable from baseline behaviour. With simulations, we demonstrate that our lasso method effectively shrinks spurious threshold effects towards zero. When applied to narwhal movement data, our analysis suggests that narwhal react to vessels up to 4 kilometres away by decreasing movement persistence and spending more time in deeper waters (average maximum depth of 356m). Overall, we provide a broadly applicable framework for quantifying behavioural responses to stimuli, with applications ranging from determining reaction thresholds to disturbance to estimating the distances at which terrestrial species, such as elephants, detect water.

Conformal prediction is a framework for constructing prediction sets for machine learning models, relying solely on the exchangeability of training and test data and without requiring to specify a parametric distribution. Despite its wide applicability and popularity, its application in single-cell transcriptomics remains underexplored. This paper addresses this gap by developing an approach that leverages the rich information about cell-type relations, encoded in the graph structure of cell ontologies, to enhance the interpretability of reference-based cell-type annotation. Leveraging conformal risk control, we develop a novel conformal algorithm for graph-structured predictions and we demonstrate how incorporating graph constraints can improve the interpretation of cell-type predictions. This approach aims to generate more coherent conformal sets that align with the inherent relationships among classes, facilitating clearer and more intuitive interpretations of model predictions. Additionally, we provide a technique to address non-exchangeability, particularly when the cell-type distribution changes between training and test datasets. We implemented our method in the open-source R package scConform, available at https://bioconductor.posit.co/packages/release/bioc/html/scConform.html.

Understanding the structure of our universe and the distribution of matter is an area of active research. As cosmological surveys grow in complexity, the development of emulators to efficiently and effectively predict matter power spectra is essential. We are particularly motivated by the Mira-Titan Universe simulation suite that, for a specified cosmological parameterization (termed a "cosmology"), provides multiple response curves of various fidelities, including correlated functional realizations. Our objective is two-fold. First, we estimate the underlying matter power spectra, with appropriate uncertainty quantification (UQ), from all of the provided curves. To this end, we propose a novel Bayesian deep Gaussian process (DGP) hierarchical model which synthesizes all the simulation information to estimate the underlying matter power spectra while providing effective UQ. Our model extends previous work on Bayesian DGPs from scalar responses to correlated functional outputs. Second, we leverage our predicted power spectra from various cosmologies in order to accurately predict the entire matter power spectra for an unobserved cosmology. For this task, we use basis function representations of the functional spectra to train a separate Gaussian process emulator. Our method performs well in synthetic exercises and against the benchmark cosmological emulator (Cosmic Emu).

The corpus callosum, the largest white matter structure in the brain, plays a critical role in interhemispheric communication. Variations in its morphology are associated with various neurological and psychological conditions, making it a key focus in neurogenetics. Age is known to influence the structure and morphology of the corpus callosum significantly, complicating the identification of specific genetic factors that contribute to its shape and size. We propose a conditional strong independence screening method to address these challenges for ultrahigh-dimensional predictors and non-Euclidean responses. Our approach incorporates prior knowledge, such as age. It introduces a novel concept of conditional metric dependence, quantifying non-linear conditional dependencies among random objects in metric spaces without relying on predefined models. We apply this framework to identify genetic factors associated with the morphology of the corpus callosum. Simulation results demonstrate the efficacy of this method across various non-Euclidean data types, highlighting its potential to drive genetic discovery in neuroscience.

This paper develops a methodological framework for reverse stress testing (RST) in which a multivariate stress scenario, coherent with the empirical dependence structure of a market, is reconstructed from a single exogenous shock prescribed on one asset class. The problem is formulated as the maximisation of the conditional density given the imposed shock, and is solved under three progressively weaker distributional assumptions. In the parametric setting, joint Gaussianity of the returns yields a closed-form modal scenario coinciding with the conditional mean of the non-shocked components. In the semiparametric setting, the modal scenario is estimated nonparametrically through the empirical likelihood methodology and the surrounding stressed trajectories are generated via a Gaussian or Student-t local sampling scheme. In the fully nonparametric setting, stressed trajectories are obtained by inverse-distance resampling of the historical observations within a Mahalanobis neighbourhood of the estimated scenario. The three variants are validated on real market data. The simulated scenarios prove to be economically coherent and capable of reproducing the standard risk-reward asymmetry observed in stressed market regimes.

Bühlmann--Straub (B-S) credibility assigns each account a weight $Z_i = E_i/(E_i+K)$, where $K$ is a single portfolio-wide ratio. The formula assumes $K$ is the same for every account regardless of size, history length, or volatility, and that recent and older years carry equal weight. On a held-out US commercial auto dataset these assumptions fail: standard B-S applied to 96 companies produces a calibration slope of 29 for small accounts, a signature of severe under-crediting. We propose a joint framework that retains B-S interpretability while addressing these limitations. The credibility weight $Z_i$ is modelled as a logistic function of account characteristics; historical experience is discounted by an EWMA decay parameter $λ$ estimated from the data; and $Z$, $λ$, and the complement are optimised in a single likelihood pass. The framework formally nests Bühlmann--Straub as a special case, admitting a likelihood-ratio test for any proposed extension. On a two-year held-out test set the proposed model restores calibration (slope = 1.00) and reduces exposure-weighted prediction error by 38% (90% bootstrap interval: 26%--50%). A size gradient in the decay rate emerges ($\hatλ\approx 0.6$, $0.84$, $0.13$ for Small, Mid, Large) and replicates qualitatively on Other Liability. A simulation study confirms the mechanisms. The model requires only account-year summaries and delivers three transparent outputs: credibility weight, complement, and recommended renewal rate.

Reliable measurement of income and consumption is essential for monitoring poverty and inequality in low- and middle-income countries, yet full household surveys are costly and difficult to implement regularly. This paper examines whether reduced survey instruments can preserve key distributional information. We apply Random Forest Recursive Feature Elimination (RF-RFE) to the 2018/19 Nigeria General Household Survey-Panel to identify the income sources, consumption categories and household characteristics that best classify individuals within the welfare distribution. The analysis focuses on three outcomes: poverty status, location in the quintile distribution and position relative to the Gini-based inequality line. The survey's post-planting and post-harvest periods allow us to assess performance under different seasonal contexts. Results show that RF-RFE achieves strong classification accuracy with few predictors. For consumption, poverty status and inequality-line position are accurately predicted using a small set of expenditure categories, while quintile classification reaches about 80 percent accuracy for seasonal consumption and 60--65 percent for annual consumption predicted from a single seasonal visit. For income, poverty status reaches around 90 percent accuracy with five predictors, and inequality-line position is largely captured by labour earnings. The findings suggest that machine-learning methods can help improve survey design and reduce data requirements while retaining much of the distributional information needed to measure and monitor poverty and inequality.

Despite Japan being one of the world's largest advanced democracies, the development of election forecasting models for its national elections remains limited. This study introduces nonlinear machine-learning forecasting models, based on decision tree and ensemble learning methods, for predicting the outcomes of Japanese lower-house elections. To assess the methodological benefits of our approach, we replicated the theoretical framework and dataset of Lewis-Beck and Tien's (LBT) foundational statistical forecasting model for Japanese elections. Our models demonstrated moderately but consistently improved predictive accuracy compared to LBT's model in both in-sample and out-of-sample evaluations, suggesting that nonlinear algorithms offer an alternative approach to classical linear methods in capturing complex electoral dynamics. This study represents one of the earlier applications of nonlinear machine-learning techniques to single-country election forecasting. It offers a replicable framework that, when combined with the country-specific electoral theories of other nations, may enhance the predictive performance of forecasting models in broader national contexts.

This study analyses the financial resilience of agricultural and food production companies in Spain amid the Ukraine-Russia war using cluster analysis based on financial ratios. This research utilizes centred log-ratios to transform financial ratios for compositional data analysis. The dataset comprises financial information from 1197 firms in Spain's agricultural and food sectors over the period 2021-2023. The analysis reveals distinct clusters of firms with varying financial performance, characterized by metrics of solvency and profitability. The results highlight an increase in resilient firms by 2023, underscoring sectoral adaptation to the conflict's economic challenges. These findings together provide insights for stakeholders and policymakers to improve sectorial stability and strategic planning.

Using standard financial ratios as variables in statistical analyses has been related to several serious problems, such as extreme outliers, asymmetry, non-normality, and non-linearity. The compositional-data methodology has been successfully applied to solve these problems and has always yielded substantially different results when compared to standard financial ratios. An under-researched area is the use of financial log-ratios computed with the compositional-data methodology to predict bankruptcy or the related terms of business default, insolvency or failure. Another under-researched area is the use of machine learning methods in combination with compositional log-ratios. The present article adapts the classical Altman bankruptcy prediction model and some of its extensions to the compositional methodology with pairwise log-ratios and three common statistical and machine learning tools: logistic regression models, k-nearest neighbours, and random forests, and compares the results with standard financial ratios. Data from the sector in the Spanish economy with the largest number of bankrupt firms according to the first two digits of the NACE code (46XX "wholesale trade, except of motor vehicles and motorcycles") were obtained from the Iberian Balance sheet Analysis System. The sample size (31,131 firms, of which 97 were bankrupt) was divided into a training and a validation dataset. The training dataset was downsampled to one healthy firm to each bankrupt firm. No outliers were removed. Focusing on predictive performance, the results show that compositional methods are better than standard ratios in terms of sensitivity (recall), with mixed results regarding specificity, compositional random forests and compositional logistic regression behaving the best.

Age-Period-Cohort (APC) models are of special importance in Demography and Epidemiology for analyzing panel data according to three different factors: biological (age), technological (period) and cultural (cohort). The main goal of APC modeling is to separate the explanation of both period and cohort effects to the phenomenon. The objective of this paper is to develop a Bayesian Age-Period-Cohort framework that can model a wide range of demographic and epidemiological phenomena and improve upon existing statistical methodologies. The APC framework consists of addressing three main challenges: (1) the identification problem of all APC models, usually managed by imposing constraints on effect groups, (2) considering expert knowledge in the model definition, and (3) efficient solution of computational issues. By allowing full parameter uncertainty, use of robust priors, and an efficient computational implementation, a Bayesian methodology manages these concerns. Bayesian models also produce results that allow intuitive implementation and support theoretical knowledge. Our original methodology consists of the use of (i) a Scaled Beta2 prior distribution for the scale parameters, (ii) imposing different period and cohort constraints and comparing them,(iii) user-friendly implementation that can be easily adapted to the event, and (iv) various model comparison criteria that leads to reasonable interpretation of APC effects. We examine the dramatic collapse of fertility in Puerto Rico, an application that is difficult to model due to the accelerated changes and has interesting demographic implications that challenge the predominance of period effects in lowest-low fertility countries, emphasizing the cohort (cultural) momentum. The scope of the methodology introduced here is wide, including applications to obesity or smoking studies, for example.

When can interventions in markets be designed to increase surplus robustly -- i.e., with high probability -- accounting for uncertainty due to imprecise information about economic primitives? In a setting with many strategic firms, each possessing some market power, we present conditions for such interventions to exist. The key condition, recoverable structure, requires large-scale complementarities among families of products. The analysis works by decomposing the incidence of interventions in terms of principal components of a Slutsky matrix. Under recoverable structure, a noisy signal of this matrix reveals enough about these principal components to design robust interventions. Our results demonstrate the usefulness of spectral methods for analyzing imperfectly observed strategic interactions with many agents.

This paper introduces a novel ranking of statistical experiments, the linear-Blackwell (LB) order, which can equivalently be characterized by (i) the dispersion of the induced posterior and likelihood ratios in the sense of the linear convex order, (ii) the size of the Lorenz zonoid (the set of statewise expectation profiles), or (iii) the variability of the posterior mean. We apply the LB order to compare experiments in binary-action decision problems and in decision problems with quasi-concave payoffs, as analyzed by Kolotilin, Corrao, and Wolitzky (2025). We also use it to compare experiments in moral hazard problems, building on Holmström (1979) and Kim (1995), and in screening problems with ex post signals.

Recent work argues for using Gaussian differential privacy (GDP) to report the privacy guarantees in privacy-preserving machine learning. We provide principled mappings from pure-DP $\varepsilon$ to GDP $μ$ by matching the worst-case success of a strong-adversary membership inference attack in terms of three metrics: multiplicative advantage at fixed FPR, precision at fixed recall, and the standard privacy profile. We tabulate $μ$ values across a useful range of parameters and recommend $μ\approx \varepsilon/5$ as a conservative general-purpose conversion.

Estimating population quantities such as mean outcomes from user feedback is fundamental to platform evaluation and social science, yet feedback is often missing not at random (MNAR): users with stronger opinions are more likely to respond, so standard estimators are biased and the estimand is not identified without additional assumptions. Existing approaches typically rely on strong parametric assumptions or bespoke auxiliary variables that may be unavailable in practice. In this paper, we develop a partial identification framework in which sharp bounds on the estimand are obtained by solving a pair of linear programs whose constraints encode the observed data structure. This formulation naturally incorporates outcome predictions from pretrained models, including large language models (LLMs), as additional linear constraints that tighten the feasible set. We call these predictions weak shadow variables: they satisfy a conditional independence assumption with respect to missingness but need not meet the completeness conditions required by classical shadow-variable methods. When predictions are sufficiently informative, the bounds collapse to a point, recovering standard identification as a special case. In finite samples, to provide valid coverage of the identified set, we propose a set-expansion estimator that achieves slower-than-$\sqrt{n}$ convergence rate in the set-identified regime and the standard $\sqrt{n}$ rate under point identification. In simulations and semi-synthetic experiments on customer-service dialogues, we find that LLM predictions are often ill-conditioned for classical shadow-variable methods yet remain highly effective in our framework. They shrink identification intervals by 75--83\% while maintaining valid coverage under realistic MNAR mechanisms.

Public datasets, crucial for modern machine learning and statistical inference, often contain low-quality or contaminated samples that can harm model performance. This creates a need for principled prefiltering procedures that a data provider can apply to protect the accuracy of a range of potential downstream statistical and learning procedures simultaneously. In this work, we formalize and analyze Learner-Agnostic Robust data Prefiltering (LARP), the problem of designing prefiltering procedures with guarantees on the worst-case loss over a pre-specified set of learners. We establish the feasibility of LARP in two theoretical settings, by providing upper-bound guarantees on the worst-case loss. Our theoretical results indicate that protecting heterogeneous learner sets via LARP comes at the price of some performance loss compared to individual, learner-specific prefiltering; we call this gap the price of LARP. To assess this gap in performance, we empirically measure the price of LARP across image and tabular tasks. We further explore potential benefits of LARP from the perspective of saving on repeated data curation efforts, in a game-theoretic model where the downstream learners can split the cost of the single prefiltering.

Fairness audits are a key component of responsible machine-learning deployment. Yet, the reliability of audit recommendations under incomplete protected-label access is still poorly understood. In this work, we focused on protected-label missingness in fairness mitigation audits. We introduced a seed-calibrated stress test to separate missingness effects from seed-to-seed movement that is already present under complete labels. Across ACS/Folktables tasks, we found that positive-availability missingness usually does not move selected mitigation methods beyond the complete-label seed floor. The no-label endpoint behaves differently, exposing ERM-equivalent candidates and deterministic tie-breaking rather than a broad missingness effect. We also found that threshold optimization can turn single-axis fairness gains into above-null intersectional harm, a sharper failure pattern that appears to remain visible under random-forest validation. Overall, our results highlight that protected-label missingness should be reported with seed-null calibration, candidate-set context, and intersectional consequences before it is treated as evidence of audit fragility.

Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attention for its ability to infer governing laws directly from experimental or simulated data, especially when the underlying physics is unclear. However, the field has expanded rapidly along diverse methodological directions, particularly with the emergence of AI-based approaches, and still lacks a clear organizing perspective. In this Review, we propose a problem-oriented perspective on data-driven differential equation discovery. We first introduce a two-dimensional phase diagram of equation discoverability, where discovery problems are organized according to structural complexity and coefficient complexity. This phase diagram shows how the field has moved from the discovery of sparse equations with simple coefficients toward more complex governing laws with richer structures and more flexible parameterizations. It also clarifies why different methodological families succeed or fail in different problem settings. We then present the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. By identifying the core problems of equation discovery that persist across algorithmic variations, REO shifts the discussion from individual algorithms to the fundamental principles that determine discoverability. We connect these perspectives to applications across physics and adjacent sciences, and argue that the next challenge is not merely recovering equations, but using them to revise existing theories, distil mechanisms and form new scientific concepts.

Large language models have been widely evaluated as simulators of individual survey responses. In practice, however, fully unobserved responses are rare; the dominant problem is partial non-response. Imputation aims to restore the overall structure of a survey dataset by filling in these missing values. It has its own well-defined evaluation criteria and differs fundamentally from prediction. We propose to impute missing survey data through in-context learning (ICL). We systematically evaluate ICL design choices across different missingness mechanisms (MCAR, MAR, MNAR) on 150 opinion variables spanning 15 waves of the American Trends Panel. Compared to well-established statistical methods for data imputation like MICE PMM, our ICL approach consistently reduces absolute error across all missingness mechanisms, with the largest gains under non-random missingness (MNAR). Notably, the best-performing specification (gpt-oss-120b with 100 in-context examples) achieves near-nominal aggregate coverage (approaching the 95% level) with confidence intervals two to five times narrower than MICE PMM. We publish a Python package with an sklearn-like API to enable easy deployment of our method using local and proprietary LLMs.

Retrieval algorithms are used to estimate atmospheric concentrations of greenhouse gases (GHGs), such as carbon dioxide (CO2) and methane (CH4), by solving inverse problems from high-spectral-resolution satellite radiance measurements. However, these algorithms are computationally expensive, which makes real-time estimation at scale difficult. Machine-learning models have therefore been proposed as fast emulators of retrieval algorithms. Most existing studies, however, evaluate them only on test data from the same period as the training data. We study the stability over time of such emulators using data from the Greenhouse Gases Observing SATellite (GOSAT). We show that prediction accuracy generally deteriorates when the test period moves away from the training period. We also show that including time as an input feature substantially improves XCH4 prediction for Lasso and neural-network models. Among the methods considered, a simple Lasso model performs as well as or better than more complex methods such as neural networks, and yields more stable predictions over time. We further validate the results using the Total Carbon Column Observing Network (TCCON), a ground-based observation network. On the TCCON-matched dataset, the time-augmented Lasso achieves errors against TCCON that are comparable to the disagreement between GOSAT and TCCON for both XCO2 and XCH4.

High-Dimensional Low-Sample Size (HDLSS) tabular domains (e.g., omics) are characterized by $n \ll m$, where $n$ = number of samples, and $m$ = number of features. Such domains often exhibit strong local correlation groups, sparse cross-group dependencies, heavy-tailed non-Gaussian marginals, heteroscedastic noise, and structured missingness, making direct density learning in $\mathbb{R}^m$ ill-conditioned since $n \ll m$. We propose BSTabDiff, a block-subunit generative framework that partitions the $m$ observed features into $M$ latent blocks ($M \ll m$) and generates each block via a shared low-dimensional subunit variable, concentrating global dependence learning in the compact block-latent space $\mathbb{R}^M$ while decoding to the full feature space with copula-driven dependence, flexible per-feature marginals, and explicit missingness mechanisms. BSTabDiff supports modern deep priors on block latents, including diffusion and normalizing flows, enabling stable synthesis and controllable benchmark generation in the HDLSS regime. Empirically, BSTabDiff produces more realistic and stable high-dimensional synthetic data when compared with unstructured tabular generators on HDLSS data.

We introduce a Process Oriented Guided Inquiry Learning (POGIL)-style activity for teaching Bayesian reasoning in introductory statistics through conditional probability, Bayes' theorem, and belief updating. The activity is self-contained, uses hand-computable probabilities organized in two-way tables, and engages students in structured team roles. We evaluated the activity in four sections of an undergraduate introductory statistics course using a quasi-experimental comparison of POGIL-style and lecture-based instruction for a Bayes' theorem unit. Outcomes included student performance on Bayes' theorem final exam questions and satisfaction with instruction. We used a Bayesian bivariate generalized linear model to compare the two approaches while accounting for major type, gender, and race. The results indicated similar exam performance and similar probabilities of high satisfaction across instructional styles and demographic groups, with considerable uncertainty and no clear evidence of meaningful differences. These findings suggest that the POGIL-style activity performed comparably to lecture-based instruction for this unit while offering an active and classroom-ready way to introduce Bayesian reasoning without requiring difficult computation or simulation. We provide adaptable instructional materials and a reproducible Bayesian analytic framework for evaluating active learning innovations in introductory statistics. Our study supports the feasible inclusion of Bayesian reasoning in introductory courses and may help instructors considering active learning.

In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier--Stokes equations. While neural operators provide fast surrogates for expensive PDE solvers, they do not by themselves provide calibrated uncertainty for spatiotemporal field predictions. Our approach wraps a trained Fourier Neural Operator (FNO) with split conformal prediction and constructs the local uncertainty scale by comparing the predictions of two operators trained on nearly identical datasets: one on the original labels and one on labels perturbed by small Gaussian noise. We consider this procedure in the data-scarce regime, where the total label budget is fixed and methods that require a separate uncertainty network must divide training data between multiple models. On the 2D Navier--Stokes benchmark, the perturbation-based method produces substantially narrower conformal bands than existing methods under matched total data budgets while maintaining the target simultaneous coverage. These results suggest that perturbation sensitivity is a practical and sample-efficient uncertainty proxy for conformalized neural operators.

Statistical post-processing has proven to be an effective tool in improving ensemble forecast of different weather variables. Case studies show that post-processing can remedy the typically underdispersive and potentially biased behaviour of the ensemble while optimizing a proper scoring rule expressing the forecast skill. The price of these positive effects is generally a deterioration in sharpness; the width of the central prediction intervals and the uncertainty of the predictions are increasing, especially for shorter lead times. This work aims to reduce the extent of the latter phenomenon for neural network-based parametric post-processing methods by extending the network's loss function with a penalty term. We demonstrate the effect of the proposed technique for 2m temperature ensemble forecasts of the European Centre for Medium-Range Weather Forecasts downloaded from the EUPPBench benchmark dataset and verified against synoptic observations. Here, the predictive distribution is Gaussian, and we use the continuous ranked probability score (CRPS) as loss function. The case studies confirm a substantial relative decrease ($8.2\%-12.5\%$) in the width of the nominal central prediction interval compared to the width of the predictive distribution computed without the penalty term, while there is no deterioration in the mean CRPS of probabilistic forecasts and in the RMSE of the predictive mean.

National statistical offices (NSOs) produce their estimates under a single weighting system (uni-weight approach): one set of weights, independent of the variable of interest, is used to estimate multiple parameters and multiple subpopulations (domains). In this paper we study, within the family of model-assisted estimators and from a design-based perspective of direct estimation, the use of regression trees as the assisting model for estimating totals in unplanned domains. We distinguish two strategies: (i) fitting a single tree at the population level and deriving from it uni-weight weights applicable to any domain, and fitting a domain-specific tree. We show that both estimators can be written as weighted sums with weights that do not depend on $y$, preserving the uni-weight property and additivity benchmarking with respect to the population total. Extending to trees the classical result, we argue why the estimator built from a population-level model tends to behave like the Horvitz-Thompson estimator within domains, whereas the domain-specific model can achieve substantial variance reductions. A simulation study based on microdata from the Uruguayan Continuous Household Survey (ECH) illustrates the behavior of the estimators at the population level and by department

Scientific progress is traditionally narrated through the interplay of theoretical insights and experimental findings. Yet this view of science underplays a third and central pillar of progress: the methods that underlie both conceptual advances and empirical evidence. By analysing more than 3 million articles across science published between 1980 and 2019, we find that science has undergone a fundamental structural transition. The share of papers that primarily contribute new methods-methods papers-has doubled across science over the past four decades, rising universally across disciplines and citation impact levels. Rather than a gradual evolution, this transition marks a pivotal shift beginning in the early 1990s, aligning with the computational revolution and the emergence of data-intensive science. The surge in methodological research is not confined to the most cited, elite publications; it spans the full spectrum of scientific output. These findings reveal a systemic reorientation of the scientific ecosystem where reusable methods increasingly serve as the essential infrastructure of scientific advances, challenging the traditional dichotomy of theory and experimental research. As science becomes increasingly methods-driven, our results call for rethinking how research is evaluated, funded and organised-towards better incentivising method innovations. This is especially the case as expanding AI must be effectively integrated with scientific instruments to realise its full potential.

Scientific machine learning is limited less by model size than by the data it is trained on. Observational data records what happened but not why; template synthetic data has a known generating process but only for the simulator's template, not the case a user faces. We argue a third option is now operationally feasible: instrumented data, in which every datum carries the mechanistic model that produced it, an explicit uncertainty over that model, and an executable family of counterfactuals. Verification-and-validation (V&V) instrumented image-to-simulation pipelines are one realisation: a sensor observation becomes a fully specified, solver-backed simulation with explicit, editable parameters and a propagated aleatoric/epistemic uncertainty. The substrate is case-specific, mechanistically supervised, and supports causal interventions through Pearl's do-operator. Near-term consequences for validation, auditing, and surrogate training span computational biology, climate, materials, fluid mechanics, and medical imaging; a longer-term, falsifiable implication concerns foundation models for scientific reasoning.

Data stream mining is fundamentally challenged by concept drift, where distributional changes can degrade model performance. Despite the proliferation of drift detection methods, progress in the field is hindered by inconsistent evaluation practices: studies rely on oversimplified synthetic data generators, adopt incompatible metrics, and lack transparency in hyperparameter selection, making fair comparisons difficult. We address this gap with a novel benchmarking framework comprising three contributions: (1) a drift simulation method that injects controlled distributional changes into real-world datasets via Monte Carlo trials, enabling supervised evaluation while preserving real-world data complexity; (2) an evaluation protocol for drift detection with timing-aware criteria, including the derivation of new metrics (e.g., F1 detection score, normalized detection time) that are comparable across streams; and (3) we advocate for a leave-one-dataset-out hyperparameter optimization protocol for drift detection methods that promotes configuration robustness across heterogeneous stream dynamics. We benchmark 14 widely used drift detection methods on 7 realworld datasets across 4 drift types (class prior, label swap, feature permutation, feature filtering), each under both abrupt and gradual transitions. Our experimental results provide insights into the strengths and weaknesses of current drift detection approaches while establishing baseline performance metrics for future research in this area. All code and experiments are publicly available.

Accurate passenger queue forecasting in airport terminals is essential for efficient departure operations, as it enables proactive congestion management. However, time-varying passenger demand and heterogeneous facility usage across multiple departure facilities make forecasting challenging. In this work, we propose a passenger queue forecasting framework that learns historical passenger flow patterns from operational data. The proposed model employs a Transformer-based architecture to capture temporal dependencies and inter-facility correlations using past queue length and waiting time at departure gates and security checkpoints, together with passenger throughput at check-in islands. The learned representations are mapped to two facility-specific MLP heads to predict queue length and waiting time at departure gates and security checkpoints. Experimental results demonstrate accurate forecasts up to two hours ahead. The proposed approach offers practical real-time decision support for proactive queue management and staff reallocation in airport terminal operations.

Accurate measurement of traffic volumes and flows is vital for modern intelligent transportation. However, despite recent technological advances in sensor devices, it is still expensive to install and maintain fixed traffic counters. Therefore, it is restricted to a small portion of location points where the counters can be installed, which severely limits the possibility of grasping and predicting the total traffic volume at a city-wide level. By contrast, devices with location history such as smartphones and connected vehicles are now widely used and provide much wider spatial coverage. However, the data from these devices are usually partial and noisy, so they are not enough to directly estimate total traffic volumes and flows. In this paper, we use the information from these widely available devices to help decide where to place additional traffic counters, and we study how selecting new measurement locations can improve city-wide traffic estimation performance. To achieve this, we propose an algorithm that chooses additional counter locations to increase the diversity of observed traffic signal patterns, rather than simply spreading counters evenly over space. The goal is to capture traffic-pattern types that are rare in the current counter set and to make the collected observations more representative for later estimation and forecasting. We also present a real-world evaluation; in a target city, we select new locations expected to improve traffic prediction, and we then commissioned new field measurements at those locations at our expense. The resulting data led to an improvement in traffic volume estimation accuracy across different fidelities.

A growing failure mode in agent evaluation and training is that models can achieve high evaluation scores by exploiting shortcuts instead of solving the intended task, producing deceptive performance. This makes evaluation scores unreliable as measures of true task-solving ability. We propose CapCode, a framework for constructing coding datasets with randomized tests whose best achievable non-cheating performance is deliberately capped below one. This capped-performance design gives evaluation scores a clearer interpretation: scores substantially above the cap are implausible and therefore provide evidence of cheating. To prevent cheating, we propose CapReward, a reward design based on the CapCode principle to discourage optimization beyond the cap. Experiments across multiple datasets show that CapCode detects cheating while preserving performance ranking of models, and CapReward reduces cheating behavior, yielding models that better follow the intended task specification.

Reliable reconstruction of missing observations in environmental panel datasets is essential for accurate exposure assessment and policy analysis. Traditional nuclear norm matrix completion methods effectively impute missing entries in low-rank matrices, yet often overlook the spatial dependence inherent to air quality processes. This paper introduces the Eigenvector Spatial Filters Nuclear Norm Matrix Completion (ESFNNMC) method, an extension of nuclear norm fixed-effects matrix completion that replaces unit-specific intercepts with a set of Moran-type eigenvectors capturing spatial autocorrelation in the data. To estimate the model, we propose a Block-Coordinate Descent (BCD) approach for multiconvex optimization problems, with soft-thresholded singular value decomposition and cross-validated regularization. Through comprehensive simulations varying missingness patterns, the level of spatial and temporal autocorrelation, and dimension, shape, and rank structure of the matrices, ESFNNMC demonstrates substantial improvements in imputation accuracy over the standard fixed-effects approach, while keeping the computational cost approximately unchanged. The method is applied to impute missing entries in daily PM10 measurements in 64 monitoring stations in Lombardy, Italy, during the year 2021.

Fitting quantitative models to data is a central step in scientific workflows, yet it remains one of the least automated. Recent agent-based systems leverage language and vision-language models (VLMs) to iteratively propose and refine statistical models, but these systems struggle on more challenging modeling tasks. To address these limitations, we introduce VESTA: Visual Exploration with Statistical Tool Agents, a framework that equips VLMs with a dynamically growing exploration toolkit to guide model refinement through data transformations, hypothesis-driven visualizations, and robust statistical tests. Unlike prior systems that rely on iterative critique alone, VESTA actively explores data before and during refinement by selecting or creating diagnostic tools, which accumulate in the model's context and can be reused later. We evaluate VESTA against established baselines in three toolkit configurations: no tools, static expert-written tools, and dynamic model-written tools. To support this evaluation, we introduce DAWN (Dataset for Automated Workflows and Numerical Modeling), a benchmark targeting distribution fitting and time series modeling with varying difficulty tiers, and culminating in real-world astronomy tasks including modeling initial mass functions and gravitational-wave chirp signals. We find that VESTA's dynamic tool creation outperforms prior agentic pipelines, with the largest gains on complex and domain-specific tasks. We further show that dynamically generated tools are substantially more sophisticated than those produced by existing visual tool-creation systems, covering more diagnostic categories per function and strongly preferring visual outputs that the VLM critic can reason over directly.

Small regional datasets pose a dual statistical problem: correlated predictors inflate estimation variance, while flexible learners can become unstable because the available information per adaptive degree of freedom is limited. We examine this issue through predictive volatility, defined as the cross-sample dispersion and upper-tail behaviour of out-of-sample loss. Using simulation evidence reported for sparse linear, near-linear and heavy-tailed settings, we compare ordinary least squares, frequentist penalties, Bayesian shrinkage models, bounded-response and spatial specifications, and flexible machine-learning procedures. In the reported simulation results, regularised linear estimators generally dominate in the linear high-collinearity micro-sample settings and remain the most reliable overall, whereas tree-based methods become more competitive only when the signal is weakly nonlinear and the sample size is larger. In the empirical application to 34 Indonesian provinces, ridge yields the best leave-one-out performance, followed by elastic net and lasso. Across the Bayesian shrinkage specifications, ICT skills show the most consistent negative association with poverty, with the strongest support under horseshoe and spike-and-slab formulations. These results suggest that, in micro-sample regional modelling, the main constraint is limited information per effective degree of freedom rather than insufficient algorithmic flexibility.

Market area models, such as the Huff model and its extensions, are widely used to estimate regional market shares and customer flows of retail and service locations. Another, now very common, area of application is the analysis of catchment areas, supply structures and the accessibility of healthcare locations. The huff Python package provides a complete workflow for market area analysis, including data import, construction of origin-destination interaction matrices, basic model analysis, parameter estimation from empirical data, calculation of distance or travel time indicators, and map visualization. Additionally, the package provides several methods of spatial accessibility analysis. The package is modular and object-oriented. It is intended for researchers in economic geography, regional economics, spatial planning, marketing, geoinformation science, and health geography. The software is openly available via the Python Package Index (PyPI) (https://pypi.org/project/huff/); its development and version history are managed in a public GitHub Repository (https://github.com/geowieland/huff_official) and archived at Zenodo (https://doi.org/10.5281/zenodo.18639559).

Deep generative models can help with data scarcity and privacy by producing synthetic training data, but they struggle in low-data, imbalanced tabular settings to fully learn the complex data distribution. We argue that striving for the full joint distribution could be overkill; for greater data efficiency, models should prioritize learning the conditional distribution $P(y\mid \bm{X})$, as suggested by recent theoretical analysis. Therefore, we overcome this limitation with \textbf{ReTabSyn}, a \textbf{Re}inforced \textbf{Tab}ular \textbf{Syn}thesis pipeline that provides direct feedback on feature correlation preservation during synthesizer training. This objective encourages the generator to prioritize the most useful predictive signals when training data is limited, thereby strengthening downstream model utility. We empirically fine-tune a language model-based generator using this approach, and across benchmarks with small sample sizes, class imbalance, and distribution shift, ReTabSyn consistently outperforms state-of-the-art baselines. Moreover, our approach can be readily extended to control various aspects of synthetic tabular data, such as applying expert-specified constraints on generated observations.

Over the last five decades, we have seen strong methodological advances in survival analysis, using parametric methods and, more prominently, methods based on non-/semi-parametric estimation. As the methodological landscape continues to evolve, the task of navigating through the multitude of methods and identifying available software resources is becoming increasingly challenging -- especially in more complex scenarios, such as when dealing with interval-censored or clustered survival data, non-proportional hazards, or dependent censoring. This tutorial explores the potential of using the framework of smooth transformation models for survival analysis in the R system for statistical computing. This framework provides a unified maximum-likelihood approach that covers a wide range of survival models, including well-established ones such as the Weibull model and a fully parametric version of the famous Cox proportional hazards model, and various extensions for more complex scenarios. We explore models for non-proportional/crossing hazards, dependent censoring, clustered observations and extensions towards personalised medicine within this framework. Using survival data from a two-arm randomised controlled trial on rectal cancer therapy, we demonstrate how survival analysis tasks can be seamlessly navigated in R within this framework using the implementation provided by the "tram" package, and few related packages.

We study an online version of the robust mean change point detection problem under a dynamic Huber contamination model with arbitrary contamination distribution and inlier distribution possessing exponentially- or polynomially-decaying tails. This robustness framework is systematically studied for the first time in the change point literature. For univariate data, we characterise the detection delay by partitioning the parameter space into four regimes, in terms of the true change location, signal size and contamination level. Efficient detection procedures are accompanied by matching lower bounds, up to poly-logarithmic factors. For the multivariate setting, we devise an efficient robust mean testing procedure and apply this to the robust online change point problem. The theoretical analysis of the robust mean testing procedure is the first in dealing with both Huber contamination and heavy-tailedness, and is thus of independent interest. Extensive numerical experiments are conducted to support our theoretical findings.

In the sense of Baire categories, we prove that the set of pairs of bivariate copulas that are comparable -- in either direction -- under the increasing convex order is nowhere dense in the space of all pairs of bivariate copulas equipped with the uniform metric. As a consequence, a topologically generic pair of bivariate copulas is not comparable under this order. We further extend the Baire-category programme to two additional stochastic orders on the space of bivariate copulas: the bivariate convex order and the stop-loss order on the sum of the components. For each of these orders, we establish that the set of comparable pairs is closed and nowhere dense, and we show that a topologically generic pair of bivariate copulas is simultaneously incomparable in all three orders. These results complement those obtained in [F. Durante, J. Fernández-Sánchez, C. Ignazzi (2022). Baire category results for stochastic orders. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, article 188] for the lower orthant order on copulas.

Bayesian inverse problems constrained by state equations are often sampled in a full parameter-state space by penalising the residual, rather than in a reduced space where the state is eliminated. We show that these formulations are not automatically equivalent as posterior measures. For finite-dimensional discretisations of equality-constrained inverse problems, assume the state equation \(c(θ,u)=0\) has a unique solution \(u=G(θ)\) and nonsingular state Jacobian \(\D_u c\). The reduced posterior, its graph lift, and the zero-noise residual posterior are then distinct. A local change of variables shows that an uncorrected Gaussian residual penalty converges, after marginalisation over \(u\), to the reduced density multiplied by \(\abs{\det \D_u c(θ,G(θ))}^{-1}\). Thus algebraically equivalent residuals can define the same feasible set but different limiting posteriors. We derive determinant corrections for unweighted, weighted, and rescaled residual penalties that have the graph-lifted reduced posterior as their hard-constraint limit. The result separates feasibility from posterior calibration: driving the residual to zero is not sufficient for exact sampling of the graph-lifted reduced posterior unless the sampling or correction step targets the corresponding corrected density.

We investigate a generalized stochastic fractional neuronal model combining fractional dynamics with correlated stochastic inputs. The proposed framework is described by a fractional differential equation driven by a latent stochastic process with stationary increments and mean-reverting structure. This formulation allows the inclusion of both short-range and long-range dependence structures and naturally produces non-exponential relaxation phenomena. The main goal is the development of a feasible parameter estimation procedure based on discrete observations of the neuronal state process. We propose a two-step methodology. First, the parameters governing the fractional dynamics are estimated by exploiting the asymptotic behavior of Mittag-Leffler functions near the origin. Subsequently, the latent stochastic input is reconstructed through fractional differentiation techniques, allowing the estimation of the parameters governing the hidden noise dynamics. We derive quantitative error bounds for the estimators and analyze the reconstruction error of the latent process under suitable regularity assumptions on the driving noise. In particular, the interplay between the order of the fractional derivative and the Hölder regularity of the noise process naturally emerges in the stability analysis of the reconstruction procedure. Finally, simulation studies illustrate the applicability of the proposed methodology and highlight the influence of memory effects and noise regularity on the quality of statistical inference. The results support the relevance of fractional stochastic analysis for the modeling and inference of neuronal systems with memory and correlated inputs.

We study finite expected-signature information for mixed-fBm paths with Hurst indices above $1/4$. Up to level three, the only parameter-dependent expected features are the variance transform $q_θ$ and the time-ordered transform $R_θ$. We prove the scale tradeoff $2K$ level-two scales versus $K$ selected level-two/three scales, together with separation and local inverse bounds.

We study finite-state finite-action Bayesian statistical decision problems. While exact error-exponent characterizations are known for several special cases, including hypothesis testing and hypothesis exclusion, the asymptotic behavior of the optimal Bayes regret is largely unknown for general decision problems. In this paper, we show that the optimal regret always decays exponentially fast and characterize its exact exponent for arbitrary loss functions. The exponent is given by the minimum multivariate Chernoff information over the minimal incompatible subsets of states, where an incompatible subset is a collection of states for which no single action is optimal for all states in the subset. Our result recovers the classical pairwise-minimum Chernoff exponent for symmetric multiple hypothesis testing and the multivariate Chernoff exponent for hypothesis exclusion, while also yielding, to the best of our knowledge, the first exact exponent characterization for list hypothesis testing.

The prediction-powered inference (PPI) proposed by Angelopoulos et al. (2023) is a popular method that leverages a small number of labeled samples and machine learning predictions for semi-supervised inference. While several variants of PPI have appeared in the literature, its rigorous statistical theory has not been fully developed. In this paper, we study the statistical optimality of PPI. Our contributions span both foundational theory and new methodology. First, we frame PPI as an M-estimation problem, revealing a link between the bias-corrected PPI estimating equation and the ideal full-data estimating equation. This connection leads to the consistency and asymptotic normality of the PPI estimator under simple random sampling without replacement. Next, we identify the efficient influence function and prove that PPI can attain the semiparametric efficiency lower bound when the predictor is score-calibrated, that is, when the predictor's output aligns with the true conditional expectation of the estimating function. Finally, for learned prediction rules, we develop asymptotic theory for cross-fitting and for a single-fit variant with variance correction in the special case of semiparametric mean estimation. Simulation experiments and a real-data application support these findings.

Optimal subsampling efficiently selects the most informative data points, enabling accurate statistical inference while significantly reducing computational burden for massive datasets. However, the existing relevant methods can not directly be applied to pairwise loss problems, particularly for convoluted rank regression (CRR), due to the double summation structure in objective function. To this end, we first propose a new BIweighted Poisson Subsampling (BIPS) framework for such problems through designing a proper weight for a pair of observations instead of for a single observation for objective function. Two concrete inverse probability weighting strategies are considered. Secondly, we focus on the CRR models, under which the BIPS estimator (BIPS-CRR) is formulated. We establish consistency and asymptotic normality for BIPS-CRR, derive its optimal Poisson subsampling probabilities under the L-optimality criterion, and provide a practical algorithm to facilitate implementation. Thirdly, we develop a distributed estimator for CRR that incorporates BIPS as a pilot subsampling strategy. This estimation is globally efficient and is robust to both randomly and non-randomly distributed datasets in distributed computing environments. Extensive simulations and a real-world application demonstrate the excellent finite-sample performance of proposed methodology. Additionally, our BIPS can be readily extended to other U-statistics optimization problems and pairwise learning tasks.

We analytically evaluate the efficiency of continuous dynamical decoupling (CDD) to curb decoherence in generic qubit setups where diverse sources of noise can be present. Previous theoretical approaches to CDD have mainly focused on its potential to cope with longitudinal fluctuations. Here, the basic scenario tackled with CDD is generalized. Apart from dealing with pure dephasing induced by diagonal noise, we consider the impact of transverse fluctuations, usually present in the practical arrangements. In particular, the implications of anisotropic noisy inputs are studied. Additionally, we analyze the role of the fluctuations in the dressing of the qubit by the CDD field of control: since the driving field is usually switched on through linear ramps of its characteristic parameters, the associated dressing of the original states can be described in terms of noisy Landau-Zener transitions. In our approach, based on a sequence of unitary transformations, the noise entering the system is cast into effective stochastic terms whose spectral characteristics are dependent on the driving parameters. This description allows the design of strategies to mitigate the impact of the fluctuations using controlled changes in the effective-noise properties. Significant robustness of CDD against the generalization of the basic scenario can be achieved through an appropriate choice of the parameters of control.

We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. Classical generic chaining characterizes the scalar quantity $\mathbb E\sup_{x\in T}X_x$; the theorem here gives a simultaneous high-probability envelope for the entire field. For an ambient prior $μ$, the envelope at $x$ is governed by a pointwise Fernique-Talagrand functional \[Φ_μ(x):=\int_0^{4σ(x)}\sqrt{\log\frac{1}{μ(B_d(x,\varepsilon))}}\,d\varepsilon,\] together with the corresponding Gaussian tail term. The theorem provides a reusable field-level refinement of classical generic chaining and a Gaussian-process counterpart of pointwise empirical-process bounds for deep neural networks. We also record a Bayesian algorithmic lower envelope from the interactive Fano/data-processing principle. For a known prior $π$, an observation channel, and a concrete estimator $\widehat t(Y)$, the lower bound is expressed through the exact ghost small-ball mass $\mathbb E_{Y\sim Q}π(B_d(\widehat t(Y),Δ))$, rather than a worst-case covering number. In Gaussian location experiments, comparison decoders convert Bayes location error into lower bounds on decision-aligned Gaussian ranges. We then construct an elementary weighted-basis example separating the usual Fano relaxation for a fixed prior, the Bayesian algorithmic lower envelope, the pointwise Gaussian envelope on the selected subatlas, and the full-class minimax risk/global Gaussian scale. Together, these results show that algorithmic lower bounds provide local-geometric certificates of pointwise complexity for fixed estimators in overparameterized ambient classes, precisely in regimes where classical minimax theory becomes either too coarse or oracle-dependent.

We begin by presenting the mathematical rationale underlying classical deductive inference. We then introduce the foundational ideas of the Bayesian inference framework. Results lying at the interface of Statistics and Ergodic Theory are outlined, providing a theoretical framework applicable to the prediction and analysis of real-world phenomena from random data. This text is expository in nature - no new results are presented; rather, recently published results are described in a didactic manner. Throughout, we work with Hölder equilibrium measures, which encompass a substantially more general class of processes than i.i.d. ones.

The Behrens--Fisher problem concerns inference on the difference of two normal means when both variances are unknown and unequal. It is a classical example in which nuisance parameters prevent ordinary exact fixed-sample inference, and it has long served as a benchmark for the foundations of inference. We revisit it through the inferential model (IM) framework of Martin and Liu. After conditioning and regular marginalization, the exact association is two-dimensional, with one coordinate for the standardized mean contrast and one for the variance ratio. Their one-dimensional generalized marginal IM is then best understood as a cylindrical two-dimensional predictive random set: sharp in its mean-contrast projection, by Hsu's stochastic domination, and vacuous in the variance ratio. Our main result is a precise validity-first optimality: among prior-free procedures that retain exact, uniform, finite-sample validity, the IM interval is the shortest. We prove minimaxity and admissibility in the cylindrical class and, by a projection argument, extend this to rectangular and general two-dimensional predictive random sets. A companion tradeoff principle shows that any adaptive procedure can only redistribute interval width across variance-ratio regimes, never shorten it uniformly. A Monte Carlo study bears this out: Welch and the bootstrap under-cover, whereas the conservative fiducial does not dominate the IM interval, being shorter only where the latter over-covers and longer where validity binds.

Diffusion-based generative models have achieved remarkable success in high-dimensional data generation; however, they fundamentally rely on isotropic diffusion processes that destroy meaningful geometric structures in the forward process. For complex, multimodal, and highly correlated distributions such as biologically constrained genetic data, isotropic noise merges distinct modes and distorts intrinsic dependencies. This forces the reverse process to recover structure from heavily degraded signals, leading to slow convergence, mode averaging, and biologically implausible samples. To address this, we introduce the Geometry-aware Ornstein-Uhlenbeck (GOU) process, a structured drift design that embeds data geometry into forward and backward dynamics. By employing a variance-aware anisotropic drift, GOU contracts low-variance directions rapidly while preserving high-variance directions longer, maintaining key multimodal structures as stable channels over time. Crucially, we show that GOU's backward initialization error is governed by local rather than global variance. This geometry-adaptive initialization improves convergence rates by reducing initial mismatch and preserving cluster-level structures. Synthetic and real-world genetic experiments demonstrate that GOU significantly improves mode separation, correlation preservation, and statistical validity over standard isotropic models.

Wasserstein metrics are increasingly adopted as similarity scores for images. We consider the sensitivity of Wasserstein metrics with respect to pixel-wise additive noise when the images are treated as discrete measures on the pixel grid. We derive finite-sample expectation bounds for a Gaussian noise model. Among other results, we prove that the error in the signed 2-Wasserstein discrepancy scales with the square root of the noise standard deviation. This is favorable compared to the Euclidean metric that scales linearly, and thus provides a theoretical basis for the benefits of optimal transport distances in noisy settings. We present experiments that support our theoretical findings and point to a peculiar phenomenon where increasing the level of noise can decrease the Wasserstein distance. A case study on cryo-electron microscopy images demonstrates that the Wasserstein metric can capture the geometry of the data manifold in high noise settings even when the Euclidean metric fails.

Software operations increasingly rely on SLOs, traces, deployment specifications, and change events, yet dashboards and thresholding practices often expose share-like operational signals as separate scalar panels or baseline distances. This can create false alarms under benign redistribution and miss movement toward policy boundaries. Rasmussen's dynamic safety model motivates drift under competing pressures, but operationalizing it for software is difficult because relevant state variables (remaining margin, engineering effort, and risk/impact) are often compositional and their parts evolve. We formulate an automated, artifact-derived drift-monitor design that maps changing software artifacts into a stable compositional monitoring state: it extracts a current part inventory and policy constraints, maps telemetry to a positive composition, stabilizes splits, merges, and renames through lineage-aware canonical groups, and analyzes boundary-directed drift in log-ratio coordinates. The proposed monitor would report drift direction, step-to-boundary, balance-level attribution, and model-health indicators under architectural churn. We specify the approach, identify its zero/noise/lineage assumptions, and report a reproducible synthetic sanity check of boundary-aware drift and controlled part churn.

Statistical inference for non-stationary data is hindered by the failure of classical central limit theorems (CLTs), not least because there is no fixed Gaussian limit to converge to. To resolve this, we introduce relative weak convergence, an extension of weak convergence that compares a statistic or process to a sequence of evolving processes. Relative weak convergence retains the essential consequences of classical weak convergence and coincides with it under stationarity. Crucially, it applies in general non-stationary settings where classical weak convergence fails. We establish concrete relative CLTs for random vectors and empirical processes, along with sequential, weighted, and bootstrap variants that parallel the state-of-the-art in stationary settings. Our framework and results offer simple, plug-in replacements for classical CLTs whenever stationarity is untenable, as illustrated by applications in nonparametric trend estimation and hypothesis testing.

Many researchers apply classical statistical decision theory to evaluate treatment choices and learn optimal policies. However, because this framework relies solely on realized outcomes under chosen actions and ignores counterfactuals, it cannot assess the quality of a decision relative to feasible alternatives at the unit level, which is an important requirement in some settings. For example, in pretrial bail decisions, a judge must balance crime prevention upon release against the risk of imposing unnecessary burdens on arrestees. A central challenge in this framework is identification: since only one potential outcome is observed per unit, counterfactual risk is typically not identifiable. We show that, under strong ignorability, counterfactual risk is identifiable if and only if the loss is additive in the potential outcomes. We further demonstrate that additive counterfactual losses can yield treatment recommendations that differ from those based on standard losses when more than two treatment options are available. We show that additive counterfactual losses capture not only decision accuracy but also decision difficulty, whereas standard losses reflect accuracy alone. Finally, we introduce a symbolic linear inverse program that determines whether a given counterfactual loss yields an identifiable risk, without requiring data.

Learning recurrent connectivity that supports memory over long intrinsic timescales is a basic problem in the theory of dynamical computation. While continuous attractor and integrator models describe how tuned recurrent circuits can maintain information, less is known about how such slow modes are acquired by gradient-based learning. Here we study this question in an analytically tractable setting: we build a mathematical theory of the learning dynamics of linear RNNs trained to integrate white noise. We show that when the initial recurrent weights are small, the dynamics of learning are described by a low-dimensional system that tracks a single outlier eigenvalue of the recurrent weights. This reveals the precise manner in which the long timescale associated with white noise integration is learned. We extend our analyses to RNNs learning a damped oscillatory filter, and find low-dimensional effective dynamical equations for the evolution of a conjugate pair of outlier eigenvalues. Taken together, our analyses build a rich mathematical framework for studying dynamical learning problems relevant to both machine learning and neuroscience.

We introduce a broad class of permutation invariant problems by extending the standard decision theoretic definition to allow also selective inference tasks, where the target is specified only after seeing the data. For any such problem, the minimizer of the risk at $\boldsymbolθ$ among all permutation invariant (equivariant) procedures is shown to be the Bayes rule that posits a uniform prior over all permutations of $\boldsymbolθ$. This gives an explicit form of the greatest lower bound on the risk of any sensible procedure in a wide range of problems. From a practical perspective, approximations to the exact bound are required because of its computational cost. In a specific example of estimating the parameter of a selected population, we prove that our bound coincides asymptotically with the computationally tractable bound attained by the Bayes rule which replaces the uniform prior on all permutations of $\boldsymbolθ$ by the i.i.d. prior with the same marginals. This generalizes results previously known only for the very special case of compound decision problems. The possibility of asymptotically attaining the latter bound by an empirical Bayes rule is discussed.