This note introduces FRESH (Fusion of Recent Evidence and Subject Histories), a method for incorporating population-level summary results -- published clinical trials, registry summaries, prior natural-history studies, and peer-reviewed indirect comparisons -- into predictive models trained on patient-level data. This method provides a principled means of combining both patient-level and aggregate-level data types into a unified data-efficient model for clinical decision making. FRESH assumes access to a generative model trained on patient-level data sources (e.g. clinical trial or real-world data). The method produces patient-level predictions from a re-calibrated model that matches a set of specified aggregate statistics for a target population. This can be understood as a patient-level recapitulation of the aggregate source -- with the key property that the recalibration is a minimal perturbation of the original joint distribution in a specific information-geometric sense. The resulting samples can be analyzed directly or combined into a post-training procedure to update the original generative model. This approach enables several applications where rigorously incorporating patient-level data with summary information is valuable, including (i) contextualizing single-arm trial results with respect to recent standard-of-care, (ii) clinical-trial simulations for design and probability-of-technical-success estimation, and (iii) comparative-effectiveness analyses of on-market therapies.

Dependence among marginally constrained observations can break a finite-sample barrier. To formalize this phenomenon, we introduce the \emph{minimum list entropy coupling} $H(P\|Q_1,\dots,Q_m)$, the minimum conditional entropy $H(X|Y_1,\dots,Y_m)$ over all joint distributions with prescribed discrete marginals $X\sim P$ and $Y_i\sim Q_i$. Unlike classical formulations based on independent observations, our model allows $Y_1,\dots,Y_m$ to be arbitrarily dependent while keeping each marginal fixed. This enlarged coupling space reveals a sharp dichotomy: independent observations reduce residual uncertainty exponentially, whereas dependent observations can eliminate it exactly after finitely many samples. We characterize this zero-entropy regime through necessary and sufficient conditions and give concrete structural criteria under which it occurs. In particular, under mild support assumptions, zero entropy is achieved with $O(\log(1/P_{\min}))$ observations, where $P_{\min}$ is the minimum nonzero mass of $P$. We also develop a greedy algorithm with monotone approximation guarantees for computing $H(P\|Q_1,\dots,Q_m)$. Finally, we show that the same framework formalizes finite-sample limits in distribution-matching representation learning and randomness extraction, where zero entropy corresponds to exact recovery and exact extraction.

Probability integral transforms (PITs) and empirical $p$-values are widely used to assess the calibration of predictive distributions. While exact PIT values are uniformly distributed under correct model specification, practical implementations rely on empirical estimates constructed from finite samples. We show that this estimation step fundamentally alters the statistical structure of the problem. In particular, common-sample and rolling-window implementations introduce dependence and variance distortions that invalidate classical one-sample uniformity tests. When empirical percentiles are conditioned on a shared reference sample, the resulting statistics converge towards a two-sample Kolmogorov--Smirnov regime, while rolling windows induce autocorrelation and variance suppression. Our findings indicate that treating empirical percentiles as independent uniform draws can distort statistical inference and that backtesting procedures based on PITs require revised calibration methods accounting for the underlying two-stage sampling structure.

Differential privacy changes the effective sample size governing CVaR learning. For tail mass $τ$, the privacy-relevant sample size is not $n$, but $nτ$; equivalently, the effective private tail sample size is $εnτ$. Private CVaR excess risk decomposes into ordinary tail-risk statistical error and a privacy price. This decomposition is complete for scalar estimation and finite classes: scalar estimation has rate $Θ(B \min\{1,(nτ)^{-1/2}+(εnτ)^{-1}\})$, and finite classes of size $M$ have rate $Θ(B \min\{1,\sqrt{\log(2M)/(nτ)}+\log(2M)/(εnτ)\})$. These complete rates hold under pure DP, and their lower bounds extend to approximate DP in the stated small-$δ$ regimes. For convex Lipschitz learning, modular upper and lower reductions show that the CVaR-specific privacy term necessarily scales as $1/(εnτ)$, with dimension dependence inherited from private stochastic convex optimization. Together, these results identify ordinary private learning on $Θ(nτ)$ informative tail records as the canonical hard subproblem inside private CVaR learning.

Flexible continuous-time survival modeling is critical for capturing complex time-varying hazard dynamics in high-dimensional data; however, training such models remains challenging due to the intractable integral required for likelihood estimation. We introduce QSurv, a scalable deep learning framework that enables nonparametric continuous-time modeling without relying on time discretization or restrictive distributional assumptions. We propose a training objective based on Gauss-Legendre numerical quadrature, which approximates the cumulative hazard with high-order accuracy while facilitating efficient end-to-end training via standard backpropagation. Furthermore, to effectively capture non-stationary hazard dynamics in complex architectures, we introduce time-conditioned low-rank adaptation, a mechanism that conditions general neural backbones on time by dynamically modulating weights via low-rank updates. We provide theoretical analysis establishing approximation error bounds for cumulative-hazard evaluation. Comprehensive experiments across synthetic benchmarks, large-scale real-world tabular datasets, and high-dimensional medical imaging tasks demonstrate that QSurv achieves competitive predictive performance with advantages in instantaneous hazard function estimation, enabling more interpretable characterization of time-varying risk patterns.

We develop a skew-adaptive extension of split conformal prediction for regression. The method starts from an asymmetric interval family centered at a point prediction and uses the gauge approach to deduce the conformity score induced by this family. The inverse hyperbolic sine transform of signed scaled residuals provides the training target for an additional predictive model, whose role is to learn how predictive uncertainty should tilt across the feature space. The resulting procedure preserves the finite-sample marginal validity of split conformal prediction under exchangeability, while producing intervals that adapt to both local scale and local skewness. We also develop a calibration-sample-based estimator for comparing the expected relative future width of the skew-adaptive and classical scaled-score intervals. Experiments on a variety of datasets indicate gains in prediction interval efficiency over the scaled-score construction and conformalized quantile regression, and show that the proposed estimator closely matches the corresponding average width ratio observed on the test sample.

We investigate the problem of covert quickest change detection in a Bayesian and infinite-horizon setting. A legitimate entity seeks to detect a change in the state of a discrete memoryless channel as quickly as possible by actively probing it. Simultaneously, the entity must ensure its probing remains covert from an adversary monitoring the channel for active sensing. We introduce the expected covertness budget (ECB) as an analytically tractable covertness metric that bounds from above the relative entropy between the observation sequences induced by active and passive sensing. Under constraints on both the probability of false alarm (PFA) and the ECB, we establish a second-order asymptotic converse bound on the average detection delay as the PFA constraint approaches zero, for any positive ECB constraint, explicitly quantifying the maximum square-root-order covert sensing gain possible. Furthermore, we propose an achievability scheme utilizing a constant-sensing-probability Shiryaev-type policy and show that it matches the second-order asymptotic converse. We illustrate our result with a numerical example.

For a fixed flow-based generative model under a small inference budget, sample quality can depend strongly on where the sampler spends its few function evaluations. Flow matching and Schrödinger bridges define probability paths, yet their inference grids are usually heuristic or inherited from one-endpoint diffusion. We derive a conditional-marginal entropy-rate objective for bridge-aware discretization, separating endpoint-conditioned bridge geometry from marginal flow evolution, and use it to build a training-free entropic inference-time scheduler from first principles. For Gaussian Brownian bridges this rate is closed-form and U-shaped, motivating boundary-heavy nonuniform grids. On trained two-dimensional bridge/flow models, the estimated profile recovers the predicted shape and improves 10-step ODE-Heun MMD over linear by 18.1%, with a paired 22.7% SDE-Heun improvement in the same low-NFE sweep. On EDM/CIFAR-10, the entropic time-discretization gives the best tested five-step FID (186.3 \pm 4.0 versus 200.5 \pm 2.9 for linear and 238.0 \pm 5.3 for cosine). On AlphaFlow protein generation, entropic conditional-marginal (cond-marg) scheduling shows advantage in low-NFE regimes on both CAMEO22 and ATLAS benchmarks. These results support entropy-rate scheduling as a practical low-budget allocation signal for high-dimensional bridge and flow samplers.

Neural network surrogate models have emerged as a promising approach to model solution fields for a wide variety of boundary value problems encountered in physical modeling. Stochastic problems represent an area of particularly high interest because of the potential to significantly reduce the repeated evaluation of expensive forward models via traditional numerical solvers when conducting parametric analysis. However, many studies found in the literature primarily focus on the ability of neural network surrogate models to represent deterministic samples or mean field solutions and largely overlook surrogate model performance at the tails of the distribution. The present study examines in detail the ability of neural network surrogate models to capture the full distribution of solution fields over the entire probability space, while emphasis is placed at the tails of the distribution. Serving as a canonical problem is the heat conduction equation with a highly stochastic source term, inducing extremely large variation in the thermal solution field. Comparisons are made between a classic feed-forward fully connected network and a Deep Operator Network architecture, using both data-driven and physics-informed loss functions. Results show that the worst-case prediction errors are an order of magnitude larger than the mean field error, highlighting the importance of the outlier samples. The large errors associated with extreme samples result from the networks having to extrapolate beyond the bounds of the training data. A method for identifying these samples is presented along with a discussion of potential approaches to account of their errors. Among the models considered, the fully connected neural network trained using a weak form residual loss performs best in handling these extrapolated inputs, achieving the highest prediction accuracy for the numerically produced datasets.

Recent advances in data collection technologies have led to the emergence of massive spatial datasets, with measurements obtained at millions of spatial locations. Geostatistical models typically employ Gaussian processes (GPs) to capture spatial dependence, but standard GP fitting becomes prohibitive at such scales. A promising solution is optimal subsampling, where a subset of locations is selected that optimizes a criterion. In this study, we propose a randomized exchange algorithm for subsampling (REX-SUB) which efficiently selects small subsamples that minimize prediction errors in the fitted spatial GP models. To further improve computational efficiency, we embed a scalable Vecchia approximation to the GP's joint likelihood, which takes advantage of sparsity in the precision matrix to enable fast inference on the selected subsamples. Through a simulation study and an application to a remotely sensed precipitable water dataset, we show that REX-SUB yields lower mean squared prediction errors and interval scores compared to competing subsampling strategies.

We propose a variational quantum classifier operating on high dimensional deep representations via amplitude encoding, stabilized by a learnable classical pre encoding layer.By combining normalized amplitude embeddings with bounded quantum observables, the resulting model induces a structured and smooth hypothesis class with controlled sensitivity to input variations. Model reliability is assessed using SAFE-AI metrics derived from the Cramer von Mises divergence, enabling consistent evaluation across accuracy, robustness, and explainability dimensions. Empirical results show that the proposed quantum model provides competitive predictive performance compared with strong classical baselines while exhibiting a more balanced SAFE reliability profile, with improved robustness to noise and stability under structured feature removal. These findings suggest that variational quantum circuits offer a principled mechanism for stability oriented SAFE learning in safety critical settings.

In-play football forecasting models have struggled to match the accuracy of betting exchange prices, which aggregate information from many market participants. We close this gap by combining two extensions to a Weibull accelerated failure time model: calibrating team strength parameters to Betfair Exchange prices at kick-off to capture pre-match market information, and including post-shot expected goals as a time-varying covariate to capture in-play information. The calibration approach, where we jointly fit team-strength parameters to 1X2 and over/under betting markets via squared-error minimisation, applies to any intensity-based goal arrival model and enables stronger in-play forecasting. Evaluated across 140 English Premier League matches at minute intervals, the calibrated model almost matches Betfair's classification accuracy (70.2% versus 70.6%) while retaining interpretable team-level parameters and covariate effects. A comparison with two alternative continuous-time scoring models, both calibrated to the same pre-match odds, confirms that market calibration is the dominant driver of predictive accuracy. A betting simulation against Betfair in-play odds yields a 4.5% return on investment (Sharpe ratio 5.94) over 17,458 bets, suggesting an inefficiency within in-play football markets.

Machine learning systems increasingly make life-changing decisions about individuals, such as loan approvals, hiring, and cheating detection, raising a pressing question: how can individuals respond to negative decisions made by these opaque systems? While explainable artificial intelligence (XAI) has largely focused on algorithmic recourse -- helping individuals change their features to obtain a desired outcome -- the parallel problem of algorithmic contestability -- helping individuals review and correct erroneous algorithmic decisions -- has received far less attention, despite its central ethical and legal importance. We trace this neglect to the absence of clear formal definitions and a systematic operationalization of contestability as an algorithmic problem. To address it, we propose an operational definition of contestability as a natural complement to recourse: contestability starts from the presumption that a decision may be incorrect and focuses on identifying evidence to challenge and potentially overturn it, whereas recourse assumes the decision is valid and instead provides pathways for changing it. We show that standard XAI explanations, such as counterfactuals, LIME, or Anchors, even when combined with human intuitions about decision continuity or monotonicity, reveal only errors in the neighborhood of the individual, but provide insufficient grounds for overturning the decision at hand. Going thus beyond traditional XAI, we identify three types of evidence warranting reversal according to the decision maker's own ethical standards: predictive multiplicity, incorrect feature values, and neglected overruling evidence. We argue that these render decisions normatively indefensible and thus successfully contestable. Finally, we analyze how existing EU legislation connects to our framework and argue that individuals already hold some legal rights to these forms of evidence.

We consider the problem of testing the mean of high-dimensional data when the dimension may grow without explicit rate restrictions relative to the sample size. The proposed procedure is based on the statistic V_n = n||Xn||^2, which avoids inversion of the covariance matrix and is therefore suitable for high-dimensional settings.We establish asymptotic distributional results for both fixed and increasing dimension by embedding the observations into the Hilbert space l2. Furthermore, we prove the asymptotic validity of a bootstrap approximation for the distribution of the test statistic. The resulting bootstrap test yields asymptotic level-a procedures without requiring sparsity assumptions or structural conditions on the covariance matrix. In all this, a new Central Limit Theorem in l2 is proving to be an extremely useful tool.

Expectations of multivariate functions with missing labels occur in various fields such as transfer learning and average treatment effects. Although non-parametric estimators based on nearest-neighbour matching are frequently used in this context, the existing literature assumes that the covariates live in some well-shaped compact subset of $\R^d$, with densities that are bounded away from zero. In this paper, we show that the usual rates of convergence can be achieved with minimal assumptions on the covariate supports. These assumptions are replaced with conditions on the source and target distributions, among which a measure of the tranferability between the two probability measures. We show that these conditions are general, can be applied to distributions supported on manifolds, and allow the target distribution to have a heavier tail than the source distribution. We also show that this control of the transferability is needed for any estimator to achieve good rates of convergence. Finally, applying our results to the estimation of treatment effects, we could relax the assumption that the assignment probabilities had to be bounded away from zero and one.

We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.

This paper introduces a quasi-Bayesian approach for local projection instrumental-variables (LP-IV) estimation. It builds a moment-based quasi-posterior using the generalized method of moments (GMM) objective and applies a roughness-penalty prior to smooth impulse responses over different horizons. The approach maintains the key first-order features of traditional LP-IV methods, while enhancing stability in finite samples and allowing for joint inference through simultaneous bands. Simulations indicate that this regularization decreases root mean squared error compared to standard GMM, especially at medium and longer horizons. An application to Danish electricity markets highlights the method's practical usefulness.

We study the classical problem of community recovery in stochastic block models with a fixed number of communities, with a twist: We seek algorithms that are stable with respect to node-wise changes in the graph structure, formally defined as a differential privacy constraint. The algorithms we develop are based on spectral clustering, where we introduce privacy to the community recovery pipeline in the form of directly privatizing the adjacency matrix; private PCA; private convex optimization; private low-rank matrix estimation; and private approximate subspace estimation. Straightforward applications of existing private algorithms lead to a rapid increase in the privacy parameter $ε$ in order to ensure consistent estimation under node differential privacy, in contrast with the simpler setting of edge privacy. To alleviate these issues, we develop novel algorithms based on (1) sampling from an exponential mechanism with a Lipschitz extension and (2) a general framework for constructing smooth projections from the space of undirected graphs to the space of bounded-degree graphs, which can then be combined with various edge-private algorithms. Importantly, the methods we develop are all computable in polynomial-time as a function of the number of nodes in the graph. We also develop novel lower bounds on the growth rate of $ε$ required in order to achieve consistent community estimation under node privacy. On a technical note, our paper highlights the complications that arise when analyzing private algorithms under the non-standard scaling $ε\rightarrow \infty$ and proposes some solutions. We also provide a novel application of the HGR maximal correlation from information theory in the context of accuracy amplification in PAC learning, which may be of independent interest.

We developed a tool for detecting domain shifts, namely subtle differences in the probability distributions of datasets. We identify these shifts using an algorithm designed to detect localised density anomalies in high-dimensional feature spaces. If an anomaly is present, we then identify the feature subspace in which the anomaly is most pronounced. This allows us to trace the domain shift to a small set of features, making the shift interpretable. Moreover, we provide a protocol for compensating domain shifts by extracting, from two unlabelled datasets, subsets of samples with no detectable residual distributional difference. We validate the framework on controlled 20-dimensional benchmarks with known ground truth, recovering both broad and localized shifts together with their supporting feature subspaces. We then apply it to healthy electrocardiogram (ECG) recordings represented by 782 features. In age- and sex-matched cohort comparisons differing in measurement-device composition, the method detects device-induced shifts, extracts representative subsets enriched in the imbalanced device components, and identifies ECG features associated with the acquisition contrast. These results suggest that density-shift detection and subspace attribution provide a practical framework for uncovering hidden cohort biases before downstream modelling.

High-dimensional classification problems often rely on the Lasso-penalized linear Support Vector Machines (SVMs). However, the double non-smoothness induced by the hinge loss and Lasso penalty in this model makes statistical inference challenging and impedes computational efficiency. In this paper, we propose a unified inference framework in both offline and online settings. In the offline case, by applying a convolution smoothing technique to the hinge loss, we construct a debiased estimator that eliminates the shrinkage bias, thereby building a valid confidence interval. For online streaming data, we develop a real-time estimator and inference procedure that relies only on summary statistics of historical data. Theoretically, we provide rigorous proofs for the asymptotic normality of our offline and online debiased estimators. Simulation studies and real data applications demonstrate that our methods achieve valid statistical inference and improved computational efficiency.

We introduce a class of Lévy-driven graph Ornstein-Uhlenbeck (grOU) models for edge-indexed network time series. The proposed framework extends generalized network autoregressive (GNAR) processes for edge-indexed network time series to continuous time and adapts graph Ornstein-Uhlenbeck dynamics, originally developed for node-indexed processes, to the edge-indexed setting. The model accommodates general Lévy noise and therefore captures both Brownian and jump behavior. We show that the model parameters can be estimated via a maximum-likelihood framework and derive the asymptotic properties of the estimator. We examine the finite-sample performance of the methodology through simulation studies and illustrate its practical relevance in an empirical application to high-frequency financial data. The results indicate that grOU models for edge-indexed network time series improve forecasting accuracy and reduce computational time relative to standard benchmarks while maintaining robustness through their network-based parametrization.

The correct inferential object in claims reserving is the conditional predictive distribution $p(R \mid \mathcal{D}, \hatθ)$, where $\mathcal{D}$ is the observed triangle held fixed. We refer to this as the conditioning principle. All existing bootstraps violate it by resampling functions of $\mathcal{D}$ inside the predictive loop, producing an $O(1)$ coverage error that does not vanish as the triangle grows. The Dirichlet-Gamma hierarchy admits a bootstrap that satisfies the principle exactly: $S^{IBNP}_i = X^{obs}_i (1-W_i)/W_i$ with $W_i \sim \mathrm{Beta}(c\hat{F}_{I-i}, c(1-\hat{F}_{I-i}))$ sampled directly from its predictive distribution. Only the allocation proportion $W_i$ is simulated; the observed triangle is held fixed. It thus inherits calibration from any development-proportion method (Chain-Ladder, Bornhuetter-Ferguson, Cape Cod, or other), making it model-agnostic. The coverage deficit is $O(I^{-1/2})$, independent of the number of development periods. Under compound Poisson data-generating processes the bootstrap is conservative for every $F_{I-i} \in (0,1)$: the predictive standard deviation analytically exceeds the true value by the factor $1/\sqrt{F_{I-i}}$. The ODP bootstrap violates the principle through two mechanisms in opposite directions: re-estimation inflates bootstrap variance under the ODP DGP, while missing accident-year frailty deflates it under frailty DGPs. The resulting coverage discrepancy is $Ω(1)$ regardless of $I$, providing a structural explanation for the cross-portfolio miscalibration heterogeneity documented by Meyers (2015). Chain-Ladder, Bornhuetter-Ferguson and Cape Cod emerge as credibility estimators under diffuse, informative and pooling priors respectively, with identical structure for counts and amounts. The concentration $c$ serves as a diagnostic: $\hat{c} < 30$ signals non-stationary development.

We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in Rényi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

Cross-fitting is a key ingredient in many semiparametric estimation procedures, such as double/debiased machine learning (DML), enabling valid estimation of low-dimensional targets in the presence of high-dimensional nuisance functions by enforcing out-of-sample use of nuisance predictions. crossfit is an R package that provides a general-purpose, estimator-agnostic cross-fitting engine. Users specify (i) a target functional and (ii) a directed acyclic graph (DAG) of nuisance models, with node-specific training fold widths and target-specific evaluation windows. The engine executes a reproducible schedule over folds, panels, and repetitions, returning either a scalar estimate (mode="estimate") or a cross-fitted predictor function for application to new data (mode="predict"). Beyond standard cross-fitting, crossfit implements fold-allocation modes that control how training data are shared across nuisance components, including disjoint and independence-enforcing allocations that duplicate reused nodes to reduce dependence between nuisance branches. The implementation targets simulation-heavy benchmarking and method development, with explicit and auditable schedules, defensive validation of specifications and nuisance dependencies, reuse-aware caching to avoid redundant refits, and failure isolation policies for large experiment grids. The crossfit package is available on CRAN, openly developed on GitHub under GPL-3, and is intended as a lightweight, tested foundation to prototype and empirically evaluate cross-fitted estimators with explicit control over fold geometry, dependence, and computation.

The distance dependent Chinese Restaurant Process (ddCRP) provides a flexible prior distribution for clustering observations, incorporating covariate information through pairwise distances and accommodating a rich variety of cluster structures. When cluster parameters are conjugate to the likelihood, Bayesian inference is straightforward. In the non-conjugate setting, however, inference becomes substantially more challenging due to the trans-dimensional parameter spaces that arise as cluster assignments change. We develop a reversible jump Markov chain Monte Carlo (RJMCMC) framework to address this challenge, targeting the dimension-changing nature of cluster parameter vectors when observation assignments are updated. We introduce and compare several proposal strategies for birth and death moves, including prior-based, independence, and data-driven moment-matching proposals that target regions of high posterior density. For fixed-dimensional moves, we propose a posterior resampling strategy that improves acceptance rates while maintaining computational efficiency. Through a simulation study and an application to Old Faithful eruption durations, we demonstrate moment-matched proposals offer a principled, data-driven alternative to prior-based proposals. The resulting methodology provides a general RJMCMC framework for ddCRP models with non-conjugate likelihoods, demonstrated here on both discrete and continuous observation models.

Researchers and practitioners in the field of reliability engineering and optimization frequently use active redundancy techniques to intensify the performance of systems. In this article, we study allocation strategies of non-matching active redundancies (spares) in coherent systems consisting of possibly dependent and identical components for achieving better system reliability. The dependence of the components is modeled through copulas using the distortion function. Sufficient conditions are derived to establish optimal allocation strategies for two heterogeneous active redundancies at the component or system levels. Moreover, the results are true for the component lifetimes following a general family of parametric distributions. The results guarantee the likelihood ratio (reversed hazard) ordering between the coherent systems at the component level (system level) active redundancies. Some aging properties are also established in this endeavor. Several examples are provided to demonstrate the theoretical results.

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact $d$-dimensional smooth manifolds $\mathcal{M} \subset [0,1]^D$ with $d > 2$ and $β$-Hölder densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_β(d)}n^{-(β+1)/(d+2β)})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.

Suppose we have an observed path from a point process counting event occurrences in a large population. Based on the observed path, we would like to test the null hypothesis that the conditional intensity of the point process belongs to a particular parametric family. We propose a novel approach to conducting such goodness-of-fit tests. The idea is to construct a unitary transformation of a natural parametric testing process such that it converges weakly to a ``standard'' target process, independent of the particular parametric form assumed under the null hypothesis. This transformation therefore paves the way for asymptotically distribution-free goodness-of-fit testing of parametric point processes. We demonstrate the good finite-sample performance of our approach through Monte Carlo simulations of Aalen-type survival processes, without and with censoring, mixture cure models, and software reliability models, and we illustrate its applicability with observed human lifetimes as well as real software failures.

The Chain-Ladder (CL) method remains the dominant macro-level technique for claims reserving in non-life insurance, yet its classical formulation lacks a coherent probabilistic foundation. Existing stochastic extensions-including the Mack model and the Over-Dispersed Poisson (ODP) framework-provide measures of uncertainty but rely on second-moment assumptions or quasi-likelihood variance structures without clear generative interpretations. This paper develops a Negative Binomial Chain-Ladder (NB-CL) model that embeds the CL method within a full likelihood-based framework. The key contribution is a micro-level derivation showing that the negative binomial distribution arises naturally from a Poisson-Gamma construction: claims arrive according to a Poisson process with Gamma-distributed accident-year heterogeneity, and aggregation yields negative binomial incremental counts. This derivation gives the dispersion parameter $κ$ a structural interpretation as accident-year heterogeneity, rather than an ad-hoc overdispersion adjustment. The NB-CL model generalises the Poisson Chain-Ladder model in the limit $κ\to \infty$, shares the point estimates of the ODP model while differing in its variance function (quadratic vs. linear), and unifies the Chain-Ladder family within a single probabilistic hierarchy. A parametric bootstrap procedure is developed to incorporate both process and parameter uncertainty. Simulation studies confirm near-nominal coverage under correct specification once the dispersion parameter is bias-corrected, and a controlled degradation under model misspecification. Empirical illustrations on claim count data (Australian motor bodily injury) and paid amounts (Taylor-Ashe) document both the structural reading of $κ$ and the working-approximation status of the model in the amounts case.

In regression models fitted to data from complex survey designs, sampling weights often incorporate non-essential variation, inflating variance estimates. Stabilized weights mitigate this issue by adjusting sampling weights to account for variation explained by covariates. In the context of two-phase sampling, we evaluate the performance of optimal stabilized weights and propose combining the stabilized weight estimator with generalized raking, a class of efficient design-based estimators. This combination improves efficiency by reducing unnecessary weight variation and leveraging information from auxiliary variables. We show this combination can be implemented using the standard statistical package that handles two-phase samples and generalized raking. Simulation studies demonstrate that the proposed estimator enhances precision under realistic two-phase designs, though efficiency gains may be limited in highly informative designs. The developed methods were applied to a large multinational two-phase study of Kaposi sarcoma among people living with HIV.

Uncertainty quantification is essential in safety-critical settings--from autonomous driving to aviation, finance, and health--where decisions must rely on conservative bounds rather than point estimates. Predictor-level intervals (e.g., from quantile regression, conformal prediction, variance networks, or Bayesian models) generally do not compose: adding two per-variable intervals need not yield a valid interval for their sum or preserve coverage. In aviation, Gaussian overbounding replaces complex error distributions with a conservative Gaussian whose tails dominate the truth, so conservatism propagates through linear operations. Yet classical overbounds are global, often overly conservative, and hard to adapt to feature-conditioned errors. We propose a unified learning framework that trains neural networks to produce context-aware Gaussian overbounds--mean and scale--with provable conservatism on a finite quantile grid and, under three explicit regularity assumptions, continuous-tail conservatism on a certified interval. Our overbounding loss enforces conservativeness at selected quantiles while penalizing distributional distance with a Wasserstein-style term. The learned bounds support conservative linear-combination and convolution analysis on the enforced grid, and on the certified interval when assumptions hold, while being less redundant than traditional methods. We provide a scoped analysis of discrete-to-continuous conservatism and compact-domain objective regularity, and validate on synthetic data and real-world datasets, including multipath, ionospheric, and tropospheric residual errors. Across these settings, the method yields tighter bounds while maintaining conservatism on the enforced grid and in experiments. The framework is modality-agnostic and applicable to learning systems that require conservative, feature-conditioned uncertainty estimates in dynamic environments.

Off-policy evaluation (OPE) estimates the value of a target treatment policy (e.g., a recommender system) using data collected by a different logging policy. It enables high-stakes experimentation without live deployment, yet in practice accuracy depends heavily on the logging policy used to collect data for computing the estimate. We study how to design logging policies that minimize OPE error for given target policies. We characterize a fundamental reward-coverage tradeoff: concentrating probability mass on high-reward actions reduces variance but risks missing signal on actions the target policy may take. We propose a unifying framework for logging policy design and derive optimal policies in canonical informational regimes where the target policy and reward distribution are (i) known, (ii) unknown, and (iii) partially known through priors or noisy estimates at logging time. Our results provide actionable guidance for firms choosing among multiple candidate recommendation systems. We demonstrate the importance of treatment selection when gathering data for OPE, and describe theoretically optimal approaches when this is a firm's primary objective. We also distill practical design principles for selecting logging policies when operational constraints prevent implementing the theoretical optimum.

Conformal prediction is often calibrated with a single pooled threshold, but this can hide cross-group heterogeneity in score distributions and distort group-wise coverage. We study this phenomenon through the population score distributions underlying split conformal calibration. First, we derive a conservation law and lower bound showing that pooled calibration incurs irreducible group-wise coverage distortion at a scale set by cross-group quantile heterogeneity. Second, we demonstrate that the two leading fairness definitions for conformal prediction, Equalized Coverage and Equalized Set Size, are fundamentally in tension. Third, we quantify the cost of moving between policies which treat groups separately or pool them. Experiments on synthetic and real data confirm the same bidirectional trade-off after finite-sample calibration. Our results show that, for the policy families studied here, calibration choice does not remove cross-group heterogeneity; it determines whether the resulting distortion appears in the coverage or size dimension, providing a principled lens for analyzing fairness-oriented calibration choices in practice.

Income inequality is a major contributor to health disparities, yet its effects often vary by geography and are commonly represented as compositional distributions (e.g., proportions of households across income brackets). Existing spatial regression methods struggle in this setting: they typically assume smooth spatial variation, cannot accommodate abrupt spatial heterogeneity, and lack principled treatment of compositional covariates. We propose a geographically weighted penalized compositional regression model that addresses these challenges simultaneously. Our method adopts a pairwise fusion penalty that enables detection of both contiguous and noncontiguous regional clusters with shared regression effects, thereby relaxing strong assumptions of spatial smoothness and geographic contiguity. This allows regions with similar underlying socioeconomic structures to be identified even when they are not geographically adjacent. By incorporating nonconvex penalties, such as the minimax concave penalty (MCP), the approach achieves improved estimation accuracy, interpretability, and scalability in high-dimensional spatial settings. We illustrate the method through an analysis linking U.S. income composition to chronic obstructive pulmonary disease (COPD) prevalence, revealing spatially heterogeneous associations that are obscured by conventional models. The proposed framework provides a flexible and robust tool for spatial data analysis involving compositional predictors and region-specific heterogeneity.

Deep generative models provide flexible frameworks for modeling complex, structured data such as images, videos, 3D objects, and texts. However, when applied to sequences of human skeletons, standard variational autoencoders (VAEs) often allocate substantial capacity to nuisance factors-such as camera orientation, subject scale, viewpoint, and execution speed-rather than the intrinsic geometry of shapes and their motion. We propose the Elastic Shape - Variational Autoencoder (ES-VAE), a geometry-aware generative model for skeletal trajectories that leverages the transported square-root velocity field (TSRVF) representation on Kendall's shape manifold. This representation inherently removes rigid translations, rotations, and global scaling of shapes, and temporal rate variability of sequences, isolating the underlying shape dynamics. The ES-VAE encoder maps skeletal sequences to a low-dimensional latent space incorporating the Riemannian logarithm map, while the decoder reconstructs sequences using the corresponding exponential map. We demonstrate the effectiveness of ES-VAE on two datasets. First, we analyze skeletal gait cycles to predict clinical mobility scores and classify subjects into healthy and post-stroke groups. Second, we evaluate action recognition on the NTU RGB+D dataset. Across both settings, ES-VAE consistently outperforms standard VAEs and a range of sequence modeling baselines, including temporal convolutional networks, transformers, and graph convolutional networks. More broadly, ES-VAE provides a principled framework for learning generative models of longitudinal data on pose shape manifolds, offering improved latent representation and downstream performance compared to existing deep learning approaches.

Sequential hypothesis tests are widely adopted as a principled way to perform multiple tests on data that arrives over time. In particular, researchers frequently utilize group sequential hypothesis tests (GST) to test the same hypotheses at K times or "groups" while data arrives sequentially. In this setting, many methods have been proposed to allow researchers to uniformly control type-1 error across K checks (often known as various alpha-spending budgets). Although these methods are all successfully valid in controlling uniform type-1 error, it is not clear which of these methods are optimal when trying to reject the null as soon as possible. In this paper, we directly optimize the rejection criterion in the GST setting under the same constraints of controlling type-1 and type-2 errors. We use a sample average approximation combined with mixed integer linear programming (S-MILP) approach for this problem and show how our S-MILP approach dominates classical GST procedures such as Lan-DeMets, Pocock, and O'Brien-Fleming methods. We also find that the optimal solution typically aggressively spends the alpha-budget early, shedding insight to the long-standing debate of which alpha-spending budgets are more efficient. We finally apply our optimal S-MILP approach to a recent study on acute kidney injury interventions and find our optimal S-MILP approach can reach the same statistically significant conclusion faster than the original study and other GST methods.

For a phenomenon $\boldsymbol{f}$ that is a function of $n$ factors, defined on a finite abelian group $G$, we derive its population statistics solely from its Fourier transform $\hat{\boldsymbol{f}}$. Our main result is an $m$-Coefficient/Index Annihilation Theorem: the $m$th moment of $\boldsymbol{f}$ becomes a series of terms, each with precisely $m$ Fourier coefficients --- and surprisingly, the coefficient indices in each term sum to zero under group addition. This condition acts like a filter, limiting which terms appear in the Fourier domain, and can reveal deeper relationships between the variables driving $\boldsymbol{f}$. These techniques can also be used as an analytical/design tool, or as a feasibility constraint in search algorithms. For functions defined on $\mathbb{Z}_2^n$, we show how the skew, kurtosis, etc. of a binomial distribution can be derived from the Fourier domain. Several other examples are presented.

Coherent Point Drift (CPD) is a representative probabilistic framework for unsupervised non-rigid point set registration. Its standard non-rigid M-step, however, relies on a point-indexed Gaussian-kernel system whose size grows with the number of moving points, making deformation estimation computationally heavy for large point sets and difficult to control in complexity during registration. To address these limitations, we propose Analytic-CPD, a new unsupervised non-rigid registration framework that gives CPD a structured analytic reformulation. Analytic-CPD preserves the CPD posterior correspondence layer, but lifts the M-step from point-indexed kernel displacement estimation to structured analytic mapping estimation. By coupling the Gaussian-mixture posterior mechanism of CPD with Structured Analytic Mappings (SAM), the method obtains a deformation model whose coefficient dimension is governed by the ambient dimension and analytic order rather than by the number of moving points. More importantly, deformation estimation is organized over an interpretable hierarchy of analytic function spaces, so the analytic order can be increased progressively as posterior correspondences become more reliable. We implement this idea through an increasing-degree continuation strategy with decreasing stage lengths: low-order analytic maps first stabilize the posterior correspondence structure, while higher-order modes later refine nonlinear residual deformation. Experiments on controlled model-matched, smooth model-mismatch, and registered human-shape data demonstrate the effectiveness and favorable accuracy--efficiency performance of Analytic-CPD.

This paper proposes a deep reinforcement learning (DRL)-based event-triggered controller design for networked artificial pancreas (AP) systems. Although existing DRL-based AP controllers typically assume periodic control updates, networked control systems (NCSs) require a reduction in communication frequency to achieve energy-efficient operation, which is directly tied to control updates. However, jointly learning both insulin dosing and update timing significantly increases the complexity of the learning problem. To alleviate this complexity, we develop a practical DRL-based controller design that avoids explicitly learning update timing by introducing a rule-based criterion defined by changes in blood glucose. As a result, decision-making occurs at irregular intervals, and the problem is naturally formulated as a semi-Markov decision process (SMDP), for which we extend a standard DRL algorithm. Numerical experiments demonstrate that the proposed method improves communication efficiency while maintaining control performance.

Bipartite networks, which encode interactions between two distinct types of entities, arise widely in applications and exhibit inherent asymmetry across node sets. Despite a growing literature on bipartite community detection, estimating community numbers $(K_1, K_2)$, a critical issue for bipartite network analysis, remains theoretically underdeveloped without any model selection consistency established, to our knowledge. Indeed, the inherent asymmetry and the two-dimensional parameter space with possibly drastically different $K_1$ and $K_2$ pose unique challenges that differ from unipartite cases. In particular, the candidate models may simultaneously overfit one node set while underfitting the other. To address these challenges, we propose Bipartite Cross-Validation (BCV), a penalized cross-validation framework that jointly selects $(K_1,K_2)$ in a fully data-driven manner. We establish the first model selection consistency for bipartite networks, notably accommodating the regime where the numbers of communities scale with the network size, revealing the intricate interplay between sparsity and model complexity. Simulations and real-data applications demonstrate strong finite-sample performance of BCV.

Fully Probabilistic design (FPD) is a powerful framework offering an elegant and unifying account of stochastic control, learning and decision-making. Here we introduce a generalized FPD framework, which we term as Tsallis FPD. Tsallis FPD uses Tsallis divergence in place of the Kullback-Leibler divergence that defines the standard FPD cost term. Tsallis divergence is a natural generalization of the KL divergence, rooted in non-extensive statistical mechanics and providing flexibility towards modeling stochastic processes with non-Gaussian tail behavior. After formulating Tsallis FPD, we develop a constructive proof of convergence by formulating a fixed point iteration. The construction takes the form of a double iteration scheme that performs a sequence of backwards inductions, rather than a single pass down the stages that constitutes the proven approach for classical FPD. We prove that this construction asymptotically converges to a fixed point and that this fixed point is an optimal solution to Tsallis FPD.

We consider Bayesian variable selection for binary outcomes under a probit link with a spike-and-slab prior on the regression coefficients. Motivated by the computational challenges encountered by Markov chain Monte Carlo (MCMC) samplers in high-dimensional regimes, we develop a mean-field variational Bayes approximation in which all variational factors admit closed-form updates, and the evidence lower bound is available in closed form. This, in turn, allows the development of an efficient coordinate ascent variational inference algorithm to find the optimal values of the variational parameters. The approach produces posterior inclusion probabilities and parameter estimates, enabling interpretable selection and prediction within a single framework. As shown in both simulated and real data applications, the proposed method successfully identifies the important variables and is orders of magnitude faster than MCMC, while maintaining comparable accuracy.

We study privacy amplification for differentially private model training with matrix factorization under random allocation (also known as the balls-in-bins model). Recent work by Choquette-Choo et al. (2025) proposes a sampling-based Monte Carlo approach to compute amplification parameters in this setting. However, their guarantees either only hold with some high probability or require random abstention by the mechanism. Furthermore, the required number of samples for ensuring $(ε,δ)$-DP is inversely proportional to $δ$. In contrast, we develop sampling-free bounds based on Rényi divergence and conditional composition. The former is facilitated by a dynamic programming formulation to efficiently compute the bounds. The latter complements it by offering stronger privacy guarantees for small $ε$, where Rényi divergence bounds inherently lead to an over-approximation. Our framework applies to arbitrary banded and non-banded matrices. Through numerical comparisons, we demonstrate the efficacy of our approach across a broad range of matrix mechanisms used in research and practice.

In this letter, we address the problem of developing quantum state tomography (QST) methods that remain valid at any time during a sequence of measurements. Specifically, the aim is to provide a rigorous quantification of the uncertainty associated with the current state estimate as data are acquired incrementally. To this end, the proposed framework augments existing QST techniques by associating current point estimates of the state with confidence sets that are guaranteed to contain the true quantum state with a user-defined probability. The methodology is grounded in recent statistical advances in anytime-valid confidence sequences. Numerical results confirm the theoretical coverage properties of the proposed anytime-valid QST.

In this paper, we apply the Feature Space Decomposition (FSD) method developed in [LS24, GLS25, LSSW26, ALSS26] to obtain, under fairly general conditions, matching upper and lower bounds for the population excess risk of spectral methods in linear regression under the squared loss, for every covariance and every signal. This result enables us, for a given linear regression problem, to define a pre-order on the set of spectral methods according to their convergence rates, thereby characterizing which spectral algorithm is superior for that specific problem. Furthermore, this allows us to generalize the saturation effect proposed in inverse problems and to provide necessary and sufficient conditions for its occurrence. Our method also shows that, under broad conditions, any spectral algorithm cannot overcome the barrier of the information exponent in problems such as single-index learning.

Quantum machine learning seeks to leverage quantum computers to improve upon classical machine learning algorithms. Currently, robust uncertainty quantification methods remain underdeveloped in the quantum domain, despite the critical need for reliable and trustworthy predictions. Recent work has introduced quantum conformal prediction, a framework that produces prediction sets that are guaranteed to contain the true outcome with a user-specified probability. In this work, we formalise how the time-varying noise inherent in quantum processors can undermine conformal guarantees, even when calibration and test data are exchangeable. To address this challenge, we draw on Adaptive Conformal Inference, a method which maintains validity over time via repeated recalibration. We introduce Adaptive Quantum Conformal Prediction (AQCP), an algorithm which provides asymptotic average coverage guarantees under arbitrary hardware noise conditions. Empirical studies on an IBM quantum processor demonstrate that AQCP achieves the target coverage level and exhibits greater stability than quantum conformal prediction.

An ongoing challenge in animal ecology is developing movement models that account for the autocorrelation, and often temporal irregularity, in telemetry data. Continuous-time Langevin diffusion models have been proposed to model temporally autocorrelated and irregularly sampled data. However, current estimation techniques obtain increasingly biased parameter estimates as the time between observations increases. In this paper, we propose using Brownian bridges in an importance sampling scheme to improve the likelihood approximation of the Langevin diffusion model. In a series of simulation studies, we showed that our approach effectively removed the bias under various scenarios. We found that the precision of the estimated habitat coefficients increased for data spanning a longer duration at a lower frequency than for shorter, more frequently sampled tracks. This suggests that the model may be well suited for modelling tracking data sampled at a coarser resolution, as is common in datasets collected with older generations of animal tags. We illustrated the application of our model using tracking data from Steller sea lions, \textit{Eumetopias jubatus}. We found that the coefficient estimates converged to values significantly different than those estimated in previous studies, suggesting that bias in conventional estimation methods may meaningfully affect ecological conclusions about habitat preference. Together, these improvements broaden the applicability of Langevin diffusion models, thereby improving ecological insight into habitat selection.

We begin our journey by recalling the fundamentals of Probability Theory that underlie one of its most significant applications to real-world problems: Parametric Estimation. Throughout the text, we systematically develop this theme by presenting and discussing the main tools it encompasses (concentration inequalities, limit theorems, confidence intervals, maximum likelihood, least squares, and hypothesis testing) always with an eye toward both their theoretical underpinnings and practical relevance. While our approach follows the broad contours of conventional expositions, we depart from tradition by consistently exploring the geometric aspects of probability, particularly the invariance properties of normally distributed random vectors. This geometric perspective is taken further in an extended appendix, where we introduce the rudiments of Brownian motion and the corresponding stochastic calculus, culminating in Itô's celebrated change-of-variables formula. To highlight its scope and elegance, we present some of its most striking applications: the sharp Gaussian concentration inequality (a central example of the "concentration of measure phenomenon"), the Feynman-Kac formula (used to derive a path integral representation for the Laplacian heat kernel), and, as a concluding delicacy, the Black-Scholes strategy in Finance.

We study a minimal change to an observation-driven Bayesian Dirichlet ARMA (B--DARMA) for compositional time series: replace the raw additive log-ratio (ALR) residual in the moving-average block with a centered innovation that subtracts the Dirichlet conditional ALR mean, available in closed form via digamma identities. We prove a recursion-level first-order equivalence (in $1/ϕ$) between the centered specification and a digamma-link DARMA at fixed parameters, under explicit interior and lag-stability conditions. The result clarifies why the two specifications should be predictively indistinguishable in the high-precision regime but does not by itself govern the geometry of the Bayesian posteriors that re-estimation produces. On weekly Federal Reserve H.8 bank-asset shares (October~2015 through October~2025, $T=522$ weeks), predictive performance is statistically indistinguishable across $104$ rolling weekly origins on every accuracy metric examined, while Hamiltonian Monte Carlo divergent transitions are approximately an order of magnitude more frequent under the raw specification, driven by isolated rolling fits at which the raw posterior exhibits localized pathologies. A four-reference sensitivity analysis confirms that predictive equivalence is reference-invariant and that the geometric advantage of centering is preserved across references but varies with the prevalence of pathological raw fits, from a substantial reduction at the loans reference to parity at the cash reference. The practical implication is operational rather than predictive: centering avoids the catastrophic raw-MA divergence spikes that occur at isolated rolling origins, which matters for production workflows in which posterior simulation feeds downstream stress tests. The adjustment is analytic and plug-in, and requires only a local change to the MA innovation calculation.

Language models (LMs) are typically post-trained for desired capabilities and behaviors via weight-based or prompt-based steering, but the former is time-consuming and expensive, and the latter is not precisely controllable and often requires manual trial-and-error. While activation steering (AS) promises a cheap, fast, and controllable alternative to the two existing post-training methods, current AS techniques require hand-crafted prompt pairs or labor-intensive feature annotation, making them more inconvenient than the plug-and-play methods such as Reinforcement Learning (RL) and Supervised Fine-Tuning (SFT). We introduce Painless Activation Steering (PAS), a family of fully automated methods that make AS readily usable with any given labeled dataset, with no need for prompt construction, feature labeling, or human intervention. We evaluate PAS on three open-weight models (Llama3.1-8B-Instruct, DeepSeek-R1-Distill-8B, and Nous-Hermes-2) and 18 tasks; we find that PAS reliably improves performance for behavior tasks, but not for intelligence-oriented tasks. The introspective variant (iPAS) delivers the strongest causal steering effects (10.1% on Bias, 5.2% on Morality, and 34.8% on Alignment). We also show PAS delivers additional gains on top of In-Context Learning (ICL) and SFT. PAS constructs a fast, lightweight activation vector that can be cheaply trained, easily stored, and activated at will. Our results provide a characterization of where AS helps, where it fails, and how to deploy it as a practical, automated LM post-training option.

We present a novel framework (TS+TT) to nest a Target Study (TS) within a Target Trial (TT) for evaluating the effects of interventions on disparities. The TS component grounds the measurement of disparity in ethical assumptions, based on the concept of allowability, and anchors it to an explicit population within calendar time. It specifies an enrollment plan of stratified sampling of eligible persons to yield a sample where social groups are distributionally similar on covariates deemed allowable for measuring disparity. Within this enrolled sample, the TT component specifies randomization of intervention strategies within each social group. Because social groups are similarly situated on allowable covariates at baseline, and because assigned intervention arms are exchangeable within social groups, TS+TT reflects a meaningful causal estimand for evaluating how interventions impact disparity. We describe the framework's key components, its emulation, and demonstrate its application to evaluate how hypothetical interventions on pulse oximeter bias affect disparities in treatment receipt in clinical care. We also extend semiparametric G-computation to accommodate continuous stochastic interventions and estimate counterfactual disparities in time-to-event outcomes. The TS+TT framework offers a versatile and policy-relevant approach for generating ethically informed causal evidence to reduce disparities and avoid exacerbating disparities.

In this paper, we derive the distribution of a two-dimensional (complex) random walk in which the angle of each step is restricted to a subset of the circle. This setting appears in various domains, such as in over-the-air computation in signal processing. In particular, we derive the exact joint and marginal distributions for two steps, numerical solutions for a general number of steps, and approximations for a large number of steps. Furthermore, we provide an exact characterization of the support for an arbitrary number of steps. The results in this work provide a reference for future work involving such problems.

One of the forthcoming major challenges in particle physics is the experimental determination of the Higgs trilinear self-coupling. While efforts have largely focused on on-shell double- and single-Higgs production in proton-proton collisions, off-shell Higgs production has also been proposed as a valuable complementary probe. In this article, we design a hybrid neural simulation-based inference (NSBI) approach to construct a likelihood of the Higgs signal incorporating modifications from the Standard Model effective field theory (SMEFT), relevant background processes, and quantum interference effects. It leverages the training efficiency of matrix-element-enhanced techniques, which are vital for robust SMEFT applications, while also incorporating the practical advantages of classification-based methods for effective background estimates. We demonstrate that our NSBI approach achieves sensitivity close to the theoretical optimum and provide expected constraints for the high-luminosity upgrade of the Large Hadron Collider. While we primarily concentrate on the Higgs trilinear self-coupling, we also consider constraints on other SMEFT operators that affect off-shell Higgs production.

Data fusion models are widely used in air quality monitoring to integrate in situ and large-scale gridded products, offering spatially complete and temporally detailed estimates. However, traditional Gaussian-based models often underestimate extreme pollution values, leading to biased risk assessments. To address this, we present a Bayesian hierarchical data fusion framework rooted in extreme value theory, using the Dirac-delta generalised Pareto distribution to jointly account for threshold and non-threshold exceedances while preserving the timing of exceedance and non-exceedance episodes. Our model is used to describe and predict censored threshold exceedances of PM2.5 pollution in the Greater London region by using CAMS atmospheric composition reanalysis, and in situ observation stations from the automatic urban and rural network (AURN) run by the UK government. Key features of our approach include combining data with varying spatio-temporal resolutions and fully accounting for parameter uncertainties. Results show that our model outperforms Gaussian-based alternatives and standalone reanalysis data in predicting threshold exceedances at the majority of observation sites and can even result in improved spatial patterns of PM2.5 pollution than those discernible from the background data. Moreover, our approach captures greater variability and spatial patterns, such as higher PM2.5 concentrations near coastal areas, which are not evident in the reanalysis data alone.

We consider the problem of modeling the effects of perturbations like gene knockouts on measurements such as single-cell RNA counts. Given data for some perturbations, we aim to predict the distribution of measurements for new combinations of perturbations. To address this challenging extrapolation task, we posit that perturbations act additively in a suitable, unknown embedding space. We formulate the data-generating process as a latent variable model, in which perturbations amount to mean shifts in latent space and can be combined additively. We then prove that, given sufficiently diverse training perturbations, the representation and perturbation effects are identifiable up to orthogonal transformation and use this to derive extrapolation guarantees for unseen perturbations that can be expressed as linear combinations of seen ones. To estimate the model from data, we propose the perturbation distribution autoencoder (PDAE), which is trained by maximizing the distributional similarity between true and simulated perturbation distributions. The trained model can then be used to predict previously unseen perturbation distributions. In support of our theoretical results, we demonstrate through simulations that PDAE can accurately predict the effects of unseen but identifiable perturbations, and showcase the method on combinatorial gene perturbation data.

Detecting localized differences between two samples is a central task in scientific data analysis, required for the identification of signal events, regime changes, or model mismatch. We introduce EagleEye, a method that pinpoints local over- and under-densities in multivariate feature spaces. EagleEye assigns each point an anomaly score by encoding its ordered k-nearest-neighbour list as a binary membership sequence and testing whether the cumulative number of successes in this sequence is consistent with a binomial (coin-flipping) null model. In the presence of a genuine local anomaly, neighbours will preferentially belong to one of the two datasts, yielding an excess of ``successes'' relative to the binomial null model. These local, pointwise detections are consolidated into interpretable anomaly sets through a deterministic refinement procedure that can also estimate the irreducible background and local density anomaly purity. We demonstrate EagleEye's efficacy in three scenarios. We first consider an artificial data example with known localized over- and under-densities. Second, we demonstrate how EagleEye may be used for new physics searches at particle collider experiments in the presence of systematic background modelling differences. Finally, we conduct a climate analysis study that reveals localized changes in spatiotemporal temperature-pattern recurrence.

Pollinators play a crucial role for plant reproduction, either in natural ecosystem or in human-modified landscape. Global change drivers,including climate change or land use modifications, can alter the plant-pollinator interactions. To assess the potential influence of global change drivers on pollination, large-scale interactions, climate and land use data are required. While recent machine learning methods, such as graph neural networks (GNNs), allow the analysis of such datasets, interpreting their results can be challenging. We explore existing methods for interpreting GNNs in order to highlight the effects of various environmental covariates on pollination network connectivity. An extensive simulation study is performed to confirm whether these methods can detect the interactive effect between a covariate and a genus of plant on connectivity, and whether the application of debiasing techniques influences the estimation of these effects. An application on the Spipoll dataset, with and without accounting for sampling effects, highlights the potential impact of land use on network connectivity and shows that accounting for sampling effects partially alters the estimation of these effects.

Distributional regression aims to find the best candidate in a given parametric family of conditional distributions to model a given dataset. As each candidate in the distribution family can be identified by the corresponding distribution parameters, a common approach for this task is to use the maximum likelihood estimator (MLE) for the parameters. In this paper, we establish theoretical results for this estimator in case the response variable is subject to random right censoring. In particular, we provide proofs of almost sure consistency and asymptotic normality of the MLE under censoring. The empirical behavior is illustrated by a simulation study and a real data example.

Predictive models often degrade in performance due to evolving data distributions, a phenomenon known as data drift. Among its forms, concept drift, where the relationship between explanatory variables and the response variable changes, is particularly challenging to detect and adapt to. Traditional drift detection methods often rely on metrics such as accuracy or marginal variable distributions, which may fail to capture subtle but important conceptual changes. This paper proposes a novel method, Profile Drift Detection (PDD), which enables both the detection of concept drift and an enhanced understanding of its underlying causes by leveraging an explainable AI tool: Partial Dependence Profiles (PDPs). PDD quantifies changes in PDPs through new drift metrics that are sensitive to shifts in the data stream while remaining computationally efficient. This approach is aligned with MLOps practices, emphasizing continuous model monitoring and adaptive retraining in dynamic environments. Experiments on synthetic and real-world datasets demonstrate that PDD outperforms existing methods by maintaining high predictive performance while effectively balancing sensitivity and stability in drift signals. The results highlight its suitability for real-time applications, and the paper concludes by discussing the method's advantages, limitations, and potential extensions to broader use cases.

Rerandomization is an effective treatment allocation procedure to control for baseline covariate imbalance. For estimating the average treatment effect, rerandomization has been previously shown to improve the precision of the unadjusted and the linearly-adjusted estimators over simple randomization without compromising consistency. However, it remains unclear whether such results apply more generally to the class of M-estimators, including the g-computation formula with generalized linear regression and doubly-robust methods, and more broadly, to efficient estimators with data-adaptive machine learners. In this paper, we develop the asymptotic theory for a more general class of covariate-adjusted estimators under rerandomization and its stratified extension. We prove that the asymptotic linearity and the influence function remain identical for any M-estimator under simple randomization and rerandomization, but rerandomization may lead to a non-Gaussian asymptotic distribution. We further explain, drawing examples from several common M-estimators, that asymptotic normality can be achieved if rerandomization variables are appropriately adjusted for in the final estimator. These results are extended to stratified rerandomization. Finally, we study the asymptotic theory for efficient estimators based on data-adaptive machine learners, and prove their efficiency optimality under rerandomization and stratified rerandomization. Our results are demonstrated via simulations and re-analyses of a cluster-randomized experiment that used stratified rerandomization.

We consider the problem of boundary detection for areal data, focusing on situations where for each areal unit multiple observations are available. We propose a Bayesian nonparametric mixture model for the area-specific population densities, with spatially dependent weights and a random number of components. Contrary to previously proposed methods for boundary detection, which consider one observation per areal unit, ours does not require external information such as area-specific covariates or dissimilarity metrics. Instead, by exploiting information from multiple samples per area, it is able to identify boundaries between areas that exhibit different densities. Crucially, the number of mixture components needs to be learned from data to obtain meaningful boundary detection, due to the non-identifiability of overfitted mixtures. Therefore, we assume it random by placing a prior on it. The motivating application is the analysis of economic inequality in the greater Los Angeles region, which typically yields social inequality and unrest. Efficient posterior computation is facilitated by a transdimensional Markov Chain Monte Carlo sampler which exploits the recently introduced optimal auxiliary priors to improve the mixing. The methodology is validated via extensive simulations and applied to the income data in the greater Los Angeles region. We identify several boundaries in the income distributions, which can be explained ex-post in terms of the percentage of the population without health insurance, though not in terms of the total number of crimes, showing the usefulness of such an analysis to policymakers.

Acute psychological stress occurs in a wide range of everyday contexts, including transportation, occupational settings, and physical activity, where its reliable detection could enable adaptive system responses and support human well-being. A persistent challenge in automated stress recognition is disentangling the biometric signatures of acute psychological stress from those of concurrent physical exertion. This study examined how five physiological signals (tonic electrodermal activity, trapezius electromyography, heart rate, heart rate variability, and respiration rate) respond to cognitive stress and physical activity, independently and in combination. Nineteen participants completed a 2x3 within-subjects design in which acute psychological stress was induced via an n-back arithmetic task combined with social pressure and financial reward, across three activity conditions: idle sitting, walking, and stationary cycling. Multilevel linear mixed models and repeated-measures ANOVA were used to decompose main effects and interactions for each sensor. Tonic electrodermal activity showed a robust, additive response to both cognitive stress (r=0.48) and physical exertion (r=0.67), with no interaction, making it the most promising candidate for stress detection during physical activity. Heart rate and trapezius electromyography were driven almost exclusively by physical exertion, with no reliable sensitivity to the stress task. RMSSD was strongly suppressed by physical activity and showed only marginal sensitivity to cognitive load. Respiration rate was dominated by physical activity, with no reliable stress effect in the primary analysis. These findings provide a sensor-specific hierarchy for real-world stress detection and highlight tonic electrodermal activity as the most informative channel when cognitive stress must be identified in physically active populations.

Causal inference with time-to-event outcomes is fundamental in various scientific studies. In a static setup with fitted propensity scores, weighted Kaplan-Meier estimation for survival probabilities and weighted Breslow-Peto estimation for hazard ratios have been widely used, but their statistical properties have been overlooked or studied only to a limited extent. We re-examine the weighted Kaplan-Meier method by formally linking it with the general framework of augmented inverse probability weighted estimation including both point and variance estimation. Furthermore, to address limitations of existing weighted methods for survival analysis, we develop new methods and associated theory through calibrated estimation in both low-dimensional and high-dimensional settings. We present a simulation study and an empirical application on the effectiveness of adjunctive psychotropic treatments for patients with schizophrenia. The calibrated methods yield coverage proportions closer to target ones in the simulation study, and produce shorter confidence intervals in both simulation and empirical studies.

We study episodic reinforcement learning with fixed reward and transition functions, but with episode-dependent admissible action sets that are observed at the start of each episode. Performance is measured by cumulative regret against the episode-wise optimal value, $\sum_{k=1}^K [V^{*,M^k} - V^{π^k,M^k}]$, where $M^k$ represents the action context in the $k$-th episode. We show that the MVP algorithm naturally extends to this framework and enjoys strong theoretical guarantees. In particular, we establish a minimax regret bound of $\widetilde{O}(\sqrt{SAH^3K\log L})$ for adversarial contexts, where $L$ denotes the number of possible contexts. This result implies a regret bound of $\widetilde{O}(\sqrt{SAH^3K})$ for stochastic contexts. We further translate the stochastic regret guarantee into a sample complexity bound of $\widetilde{O}(SAH^3/ε^2)$ for a fixed context distribution. In addition, we derive a gap-dependent regret bound of \[ \widetilde O\left( \inf_{p\in [0,1)} \left( \frac{1}{Δ_{\min}^{p}} + pKΔ_{\min}^{p} \right)\log K \cdot \mathrm{poly}(S,A,H) \right), \] where $Δ_{\min}^{p}$ is the global $p$-trimmed positive-gap floor over suboptimal $(h,s,a)$ triples. This bound can substantially improve upon the minimax rate when the relevant suboptimality gaps are large.

Concept Activation Vectors (CAVs) are a fundamental tool for concept-based explainability in deep learning, yet their practical utility is limited by statistical instability. We analyze the stochastic nature of CAVs and the Testing with CAVs (TCAV) method, deriving the distributions of major CAV classes including PatternCAV, FastCAV, and ridge regression-based CAVs. We then identify a fundamental flaw in the standard TCAV score: its reliance on a discontinuous indicator function induces non-decaying variance in critical regimes. To address this, we introduce $α$-TCAV, a generalized framework that replaces the indicator with a parameterized smooth function, yielding a unified probabilistic formulation that subsumes both TCAV and Multi-TCAV. We characterize the induced distributions of sensitivity scores and different TCAV variants, showing that established state-of-the-art choices lack theoretical justification. We provide principled guidance on tuning the parameter in $α$-TCAV -- either to imitate Multi-TCAV at substantially lower computational cost, or to obtain a calibrated Bayes-optimal probabilistic measure of a concept's influence. Finally, our analysis yields practical recommendations that challenge established routines: most notably, allocating the full sampling budget to a single CAV rather than splitting it across several.

We propose a joint order-based scoring framework for causal structure learning of directed acyclic graph (DAG) models under heterogeneous data settings. We show that leveraging heterogeneity improves the accuracy of causal ordering estimation. In the most favorable case, the causal ordering is identifiable up to two permutations. Building on this framework, we propose an order-based Bayesian method for Gaussian DAG models and establish its theoretical properties in the high-dimensional regime. For posterior inference over the space of orderings, we introduce a random-to-random (R2R) proposal neighborhood for the Metropolis-Hastings algorithm, which is theoretically motivated and exhibits efficient mixing behavior. Simulation studies confirm the strong empirical performance of the proposed method, and an application to single-nucleus RNA sequencing data from major depressive disorder demonstrates practical utility.

Background: Survival prediction models are often less reliable in clinical groups with limited sample sizes or few outcome events. Target-only models may be unstable, whereas models from larger cohorts may transfer poorly when risk-factor effects differ across populations. We evaluated whether structured transfer learning can improve survival risk stratification in data-sparse cohorts while allowing cohort-specific adaptation. Methods: We developed the COhort-shared Rank-rEduced Cox model (CORE-Cox), a two-stage framework for multi-outcome survival prediction. CORE-Cox learns shared risk-factor patterns across related outcomes in a larger source cohort via a low-rank Cox coefficient structure, then adapts these patterns to a smaller target cohort through regularized residual correction. We evaluated CORE-Cox in UK Biobank (White source, n=150,093; Asian target, n=2,534) and MIMIC-IV (White ICU source, n=15,997; Asian ICU target, n=672), comparing against target-only Cox, penalized Cox, low-rank multi-task, naive pooling, direct transfer, and single-outcome residual transfer under repeated nested cross-validation. Results: CORE-Cox achieved best or near-best discrimination across most outcomes. Mean C-index improved from 0.733 to 0.766 in UK Biobank and from 0.628 to 0.658 in MIMIC-IV, with gains in eight of nine outcomes. CORE-Cox also improved top-15% risk enrichment, with hazard-ratio estimates typically intermediate between source-only and target-only models. Discussion: CORE-Cox offers an interpretable transfer-learning framework for survival risk stratification in data-sparse cohorts, combining shared cross-outcome structure with cohort-specific adaptation. Further validation is needed before use in calibrated absolute-risk prediction or clinical decision-making.

We study risk-aware offline policy learning, aiming to learn a decision rule from logged data that is optimal under general risk criteria. This problem is crucial in high-stakes domains where online interaction is infeasible and adverse outcomes must be carefully controlled. However, existing literature on offline contextual bandits either centers on expected-reward criteria or restricts risk considerations to policy evaluation instead of optimization. In this work, we propose a unified distributional framework for optimizing Lipschitz-continuous risk functionals, a broad class of risk measures encompassing mean-variance, entropic risk, and conditional value-at-risk, among others. By developing novel empirical concentration inequalities for importance sampling-based distributional estimators, our analysis derives data-dependent suboptimality bounds with an $\tilde{\mathcal{O}}(1/\sqrt{n})$ rate, without relying on restrictive uniform overlap assumptions. This rate is minimax optimal and matches that of risk-neutral offline policy optimization, indicating that optimizing general Lipschitz risk criteria incurs no additional statistical cost relative to the expected-reward.

We formulate and study a statistical version of Katona's two-round search problem of finding at least one excellent element in a set. A population of $n$ elements is considered, where each element is independently excellent with probability $λ/n$, $λ> 0$. A subset test is noiseless: it returns positive exactly when the queried subset contains at least one excellent element. The goal is to minimize the expected number of tests subject to finding one excellent element with probability at least $1-α$, where $0<α<1$, under the restriction that testing is performed in two rounds. Unlike classical group testing, the objective is not to recover the full set of excellent elements, but only to identify one of them. We first show that success is fundamentally limited by the possibility that no excellent element exists. In the sparse Poisson regime, this imposes the necessary feasibility condition $α\ge e^{-λ}$. When the target success probability is feasible, we prove that the optimal expected number of tests grows logarithmically with the population size. The upper bound is obtained by combining an initial existence test with a second-round separating design; the lower bound follows from an information-counting argument. Numerical illustrations show the feasibility boundary and the resulting logarithmic scaling.

Prewhitening is a common approach to deal with strong autocorrelation. In this article, we propose a new approach called tail postcoloring, motivated by it. It uses parametric models to project, or color back, the neglected tail autocovariances in nonparametric estimators onto the final estimator. This approach bridges the non-parametric variance estimator and the parametric coloring model through a scaling factor. It automatically switches between these two arms using a bandwidth parameter, without the need to transform the entire dataset into residuals, as in the standard prewhitening approach. When the coloring model is well-specified, a parametric rate can be achieved. In finite samples, it is also more robust to misspecification of the coloring model compared to the whitening model in the standard approach. Besides, it avoids severe potential variance inflation or power reduction caused by the recoloring factor in the standard approach. We show that multiple parametric models can be used to construct a multiply robust tail postcolored estimator. It also naturally works for multivariate time series. A real-data example in Markov chain Monte Carlo output analysis is provided.

Estimating the number of distinct elements in a data stream is well understood when repeated elements are identical. In modern settings, however, observations are high-dimensional and noisy, so repeated instances of the same object are only approximately similar -- for example, different images of the same individual may vary significantly at the pixel level. Classical sketches such as HyperLogLog rely on consistent hash values for identical elements and break down in this regime. Recent work on robust distinct counting in general metric spaces achieves $\widetildeΘ(\sqrt{n})$ memory, which is tight in the worst case. We show that substantially improved memory guarantees are possible under geometric structure common in learned representations. We introduce MaxSketch, a simple max-linear sketch built from random Gaussian projections, and prove that it succeeds in estimating the number of distinct latent objects. Concretely, we show that under this assumption $m = \widetilde{O} (\log n / \varepsilon^2)$ random projections (and hence $\widetilde{O} (\log n/\varepsilon^2)$ memory) suffice to recover the true distinct count within a $(1+\varepsilon)$ factor. Experiments on image streams confirm that MaxSketch accurately estimates distinct counts and generalizes beyond the training regime. Our results bridge classical streaming algorithms and modern representation learning, showing how geometric structure can fundamentally reduce the complexity of distinct counting.

We derive explicit upper bounds for the number of nondegenerate critical points of a $k$-component Gaussian mixture density in $\mathbb{R}^d$, and the number of modes when the modal set is finite, together with lower bounds. By normalizing the critical-point equations by a reference component, for $k\ge2$ we get the direct Pfaffian bound \[ U_{\mathrm{het}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+2\min(d,k-1)+1\right)^{k-1}. \] For the same parameter range, an exact elimination augmented by an algebraic reciprocal variable gives the alternative bound \[ U_{\mathrm{aug}}(d,k)= 2^{\binom{k-1}{2}}(d+1)\left((2k-1)d+2k-1\right)^{k-1}. \] Thus, for $k\ge2$, the best critical-point bound is their minimum. A Morse-theoretic argument improves the corresponding finite-mode upper bound to \[ \left\lfloor \frac{\min\{U_{\mathrm{het}}(d,k),U_{\mathrm{aug}}(d,k)\}+1}{2}\right\rfloor. \] In the homoscedastic case, for $k\ge2$, the direct bound improves to \[ U_{\mathrm{hom}}(d,k)=2^{\,d+\binom{k-1}{2}}\left(d+\min(d,k-1)+1\right)^{k-1}, \] an affine-rank reduction replaces $d$ by the affine rank of the component means, and an augmented homoscedastic reduction gives the dimension-free bound \[ U_{\mathrm{aug,hom}}(k)=2^{\binom{k-1}{2}+1}(2k)^{k-1}. \] On the lower-bound side, for $d,k\ge 2$ we obtain \[ L_{\mathrm{bin}}(d,k)=k+\max_{2\le r\le \min(d,k)}\binom{k}{r}, \] together with a padding-product family that in particular implies the linear lower bound $d+k-1$, and a seed-closure principle that packages product and padding constructions. We further give explicit bounds for the number of connected components of the critical set.

Point cloud learning often rests on the premise that observed samples are noisy traces of an underlying geometric object, such as a manifold embedded in a high-dimensional feature space. Yet much of this geometry is not captured directly by coordinates, pairwise distances, or learned graph neighborhoods alone. In the smooth setting, differential forms are devices to encode higher order tangency information. In this work, we introduce a new family of principled learnable geometric features for point clouds called neural point-forms (NPFs). In the absence of a natural tangency structure, we instead use Laplacian-based techniques from Diffusion Geometry to build a discrete model for comparing differential forms on point clouds via inner products. In the continuum, submanifolds of a shared ambient feature space are represented as comparison matrices, whose entries describe how pairs of feature forms interact with extrinsic tangency information. We make this intuition precise by proving the long-run consistency of comparison matrices under standard sampling, bandwidth, density, and manifold-hypothesis assumptions. This yields a compact, efficient and permutation-invariant neural layer whose output is a learned form-comparison matrix. Across synthetic and biologically relevant experiments, we show that NPFs provide a competitive, and interpretable representation, with the strongest benefits appearing when labels depend on sampling density, manifold-like structure, or response-relevant population geometry.

Liquid-cooled exascale supercomputers dissipate heat through cooling plants organized as multiple parallel subloops, but how to allocate coolant distribution units (CDUs) across subloops and how to distribute flow among them has not been systematically addressed for facilities at this scale. This paper presents a three-layer optimization framework that jointly determines the integer partition of CDUs across subloops, the continuous flow fraction allocation, and the per-timestep co-design optimization of total flow rate and supply temperature subject to per-subloop thermal safety constraints. The Modelica simulation model is built based on the data of Frontier exascale supercomputer at Oak Ridge National Laboratory. By developing a reduced-order surrogate model, all 611 feasible partitions of 25 CDUs are evaluated across the full year operational dataset of 49,353 timesteps. Three progressively richer operational strategies are compared, ranging from flow control optimization to full three-layer co-design optimization with dynamically adjusted flow fractions. The globally optimal design is a two-subloop plant achieving 35.48% annual cooling energy savings, only 0.18% above the current three-subloop Frontier design at 35.30%. Flow fraction optimization is shown to compensate for any feasible CDU-to-subloop assignment, reducing the design sensitivity by 93% and providing a low-cost software-only pathway to near-optimal performance on the existing Frontier hardware. The framework is transferable to other liquid-cooled high-performance computing plants.

Survival analysis provides a powerful statistical framework for modeling time-to-event outcomes in the presence of censoring. However, selecting an appropriate estimator from the many specialized survival approaches often requires substantial methodological and domain expertise. We introduce SurvivalPFN, a prior-data fitted network that amortizes Bayesian inference for censored observations through in-context learning. SurvivalPFN is pretrained on a diverse family of synthetic, identifiable, and right-censored data-generating processes, enabling it to amortize survival analysis in a single forward pass during inference. As a result, the model adapts to the effective complexity of each dataset without task-specific training or hyperparameter tuning, avoids restrictive parametric assumptions, and produces calibrated survival distributions. In a large-scale benchmark spanning 61 datasets, 21 methods, and 5 evaluation metrics, SurvivalPFN achieves strong predictive performance and often improves upon established survival models. These results suggest that SurvivalPFN offers a principled and practical foundation model for survival analysis, with potential applications in high-impact domains such as healthcare, finance, and engineering (https://github.com/rgklab/SurvivalPFN).

Subgroup analyses within randomized controlled trials are often underpowered due to limited sample sizes. We address this challenge by leveraging trial participants outside the subgroup of interest to augment estimation within the subgroup. Specifically, we study two Targeted Maximum Likelihood Estimators (TMLEs) that borrow information from non-subgroup participants within the same trial: a TMLE with pooled regression (TMLE-PR) and an Adaptive Targeted Maximum Likelihood Estimator (A-TMLE). Both estimators enable information sharing without relying on any external real-world data, thereby capitalizing on key strengths of the trial: most importantly, the protection against bias afforded by the randomized treatment, but also harmonized data collection, and consistent treatment and outcome definitions. The general strategy proposed here directly advances the priorities of key regulatory agencies, including the FDA, by improving the precision of subgroup-specific treatment effect estimates without introducing external sources of bias, thereby facilitating rigorous inference to support equitable labeling, access, and post-market evaluation. In a case study based on analysis of data from a cardiovascular outcome trial (LEADER, NCT01179048), we estimate the risk reduction of major adverse cardiac events (MACE) under liraglutide treatment among Black and Asian subgroups -- each comprising less than 10\% of the trial population -- using the proposed estimators that borrow information from the remainder of the trial. Using A-TMLE, in particular, we find estimated absolute MACE risk reductions of 1.6, 1.5, and 1.5 percentage points among Asian participants and 2.1, 2.0, and 2.1 percentage points among Black participants at 365, 540, and 730 days, respectively, with 95\% confidence intervals excluding the null at each time point.

High-dimensional compositional covariates, often derived from count data, are subject to measurement error and are frequently analyzed after aggregation along a prespecified tree to improve interpretability in applications such as microbiome studies. Existing approaches typically handle either tree-guided compositional regression or errors-in-variables correction, but they do not account for the hierarchical contamination induced by their interaction. We show that tree aggregation turns leaf-level measurement error into level-dependent, correlated contamination across aggregated nodes, which inflates bias, weakens concentration rates for corrected estimating quantities, and leads to unstable variable selection for naive approaches. We propose Tree-Aggregated Regression with Correction for Observation Error (TARCO), which integrates bias-corrected estimating quantities with a tree-aware positive semidefinite stabilization and sparse regularization, with tuning selected by cross-validation based on the corrected objective. The resulting convex program can be solved with scalable algorithms. We establish finite-sample bounds for prediction and estimation errors and prove sign consistency under conditions that explicitly reflect tree heterogeneity. The guarantees persist when the measurement-error covariance is replaced by a consistent estimator. Simulations across multiple tree depths and a microbiome application demonstrate improved estimation accuracy, support recovery, and aggregation-level interpretability compared with methods that ignore the interaction between tree aggregation and measurement error.

The minima of modern neural network loss functions are typically not isolated, rather they form connected components of reparameterization invariant solutions on the training data. Analytically characterizing these solutions is a hard problem, but sampling approaches are feasible. By construction, existing methods either spread over low-loss regions, and thus do not sample reparameterization invariant solutions exactly, or are inherently local, which limits exploration of other minima valleys. We propose sampling such reparameterization invariant models using a dynamical system based on kinetic energy, subject to a gravitational pull and a friction term that dissipates energy from the system. Our proposed sampler, DiMS, is guaranteed to sample exactly from the minimum level sets and depends on physically motivated hyperparameters which allows control over the exploration capabilities of the sampler. We consider uncertainty quantification in Bayesian inference as the motivating problem and observe improved performance compared to previously proposed approaches.

Quantile regression extends regression analysis beyond the conditional mean, providing a richer characterization of covariate effects across the outcome distribution. For sensitive binary outcomes, however, misclassification due to underreporting can substantially bias inference. We propose a Bayesian quantile regression framework for misclassified binary outcomes that introduces a latent true response and explicitly models false negative and false positive reporting errors. Estimation is performed through a novel Markov chain Monte Carlo (MCMC) algorithm. Simulation studies under varying prior specifications and misclassification rates demonstrate improved performance over models that ignore misclassification. We apply the method to self-reported spousal violence data, examining associations with employment status and household wealth while adjusting for socio-demographic factors. The results indicate that underreporting exceeds overreporting across quantiles and that accounting for misclassification can change substantive conclusions.

We study contextual dynamic pricing in a semiparametric scalar-index valuation model where the latent value is $v_t=μ_\ast(\mathsf c_t)+ξ_t$, with an unknown utility map $μ_\ast$ and an unknown additive noise distribution. The key decision object is the one-dimensional oracle price map $u\mapsto p^\ast(u)$ induced by the scalar index $u=μ_\ast(\mathsf c)$ and the noise tail. Under the $β$-Hölder smoothness of the tail function for $β\geq 2$ and a revenue-geometry condition that gives a unique, stable, interior maximizer, this oracle map is itself $(β-1)$-smooth. We exploit such structure through $\mathsf{ORBIT}$, a modular coarse-to-fine policy that takes a scalar pilot index as input, localizes a benchmark price in each active bin, and learns a local polynomial approximation of the oracle map inside a trust region via bandit convex optimization. For the baseline linear utility model $μ_\ast(\mathsf c)=\mathsf c^\topθ_\ast$, an adaptive elliptical exploration scheme constructs the required scalar pilot online without distributional assumptions on the contexts. The resulting policy achieves regret $\widetilde{O}\big(T^{\frac{2β-1}{4β-3}}+\sqrt{dT}\big)$. For fixed $d$, we establish a matching lower bound in the horizon dependence, unveiling that the nonparametric oracle-map learning term is minimax sharp. The same scalar-pilot interface also yields extensions to sparse high-dimensional linear utility and nonparametric Hölder utility.

We develop a model of choice over social norms that allows for complementarities along two dimensions: \textit{technological}, analogous to complementarities between consumption goods, and social, capturing returns from conformity. Together, these determine whether two norms are complements, substitutes, or independent, as defined by how the equilibrium prevalence of one norm responds to a marginal shift in the utility of another. We estimate the model using repeated cross-sections from Sierra Leone and Nigeria, focusing on female genital cutting, polygyny, and child marriage. Social returns are significant across all specifications. For female genital cutting and child marriage, we find evidence of complementarities, especially strong in Sierra Leone. For polygyny and child marriage, we find evidence of social substitutability, particularly in Nigeria. We interpret these differences using insights from anthropology. Finally, we iterate the model forward to study policy counterfactuals, assessing the potential effects of legal reforms and social interventions.

Mixture-of-Experts (MoE) models rely on balanced expert utilization to fully realize their scalability. However, existing load-balancing methods are largely heuristic and operate on noisy mini-batch assignment statistics, introducing bias relative to population-level objectives. We propose $ϕ$-balancing, a principled framework that directly targets population-level expert balance by minimizing a strictly convex, symmetric, and differentiable potential of the expected routing distribution. Using convex duality, we derive an equivalent min-max formulation and obtain a simple online algorithm via mirror descent, yielding an efficient EMA-based routing adjustment with negligible overhead. Across large-scale pretraining and downstream fine-tuning, $ϕ$-balancing consistently outperforms prior Switch-style and loss-free baselines, demonstrating more stable and effective expert utilization.

Joint-embedding predictive architectures (JEPAs) propose that a model should learn more useful abstractions when trained to predict latent representations rather than observed outputs. For autoregressive language-model fine-tuning the principle entails a stricter requirement: the induced hidden-state geometry must reach the language-model head \emph{and} improve the decoded task metric. We test that requirement under a fixed Llama-3.2-1B-Instruct LoRA harness on natural-language-to-regex generation, comparing twenty-two training-time auxiliaries across trajectory-shape regularisation, distributional constraints, predictor/target asymmetry, Fisher-metric Jacobi residuals, and a decoder-visible JEPA objective constructed to lie in cross-entropy's positive cone. The empirical answer is a structured null: several auxiliaries clear single-cell paired $α= 0.10$ without correction (T3-Local at $Δ= +2.53$~pp, $p = 0.003$ being the strongest), but none survives Bonferroni or Holm--Bonferroni at the relevant family-wise threshold, even though many change curvature, anisotropy, variance, and gradient direction. Decoder-visible JEPA yields the first positive auxiliary--cross-entropy gradient cosine in the study, yet exact match remains inside seed noise; a full-fine-tuning replication of the same auxiliary at $n = 5$ seeds reproduces the null on both benchmarks (TURK: $Δ= +0.04$~pp, $p_{\text{paired}} = 0.96$; SYNTH: $Δ= +0.52$~pp, $p_{\text{paired}} = 0.28$), so the null is robust across LoRA and full fine-tuning for the decoder-visible construction. Hidden-state representation work and decoded-task accuracy are therefore weakly coupled in this regime; we accordingly reframe LLM-domain JEPA evaluation as a coupling problem, in which the operative question is under which metrics useful hidden geometry becomes decoder-visible task signal.

The average treatment effect can obscure important heterogeneity when individuals respond differently to a treatment. While the conditional average treatment effect (CATE) function captures such heterogeneity, it is difficult to communicate when it depends on many covariates. Sublevels sets of a multivariate CATE function are equally complicated objects, but the probability of a sublevel set of a CATE function is a single number with a simple interpretation as the proportion of individuals whose expected treatment effect does not exceed a prespecified threshold. By varying the threshold, a univariate monotone curve appears which can be used to visualize the overall type and degree of heterogeneity in a population. We formalize this curve as a target parameter and show that it is not pathwise differentiable under a nonparametric model. To address this nonstandard estimation problem, we leverage recent advances in monotone function estimation and develop a Grenander-type estimator that incorporates machine learning. We also show that the best piecewise linear approximation to the curve of interest is a pathwise differentiable parameter, and we develop a debiased machine learning estimator of this approximation. We investigate our proposed estimators' finite sample performance in a sequence of numerical studies based on data synthesized from a randomized trial. The methods are illustrated in data from a randomized trial on diabetes medication.

Accurate surrogate construction for PDE-driven high-dimensional rare-event simulation is challenging when performance evaluations are expensive. Since a globally accurate surrogate may require many high-fidelity evaluations, adaptive importance sampling provides a natural localization tool: its evolving proposal distribution progressively identifies the failure-relevant region. Motivated by this observation, we propose a surrogate-assisted adaptive importance sampling framework that refines the surrogate locally along the evolving proposal, rather than over the entire input space. The surrogate combines an encoder with a neural network, providing a low-dimensional latent representation for both prediction and sample selection. At each adaptive iteration, candidates drawn from the current proposal are selected by a greedy latent-space rule balancing proximity to the estimated failure boundary and sample diversity. The selected samples are evaluated by the high-fidelity model and used to refine the surrogate, which then guides the subsequent cross-entropy-type adaptive proposal update. We establish one-step proposal stability bounds under local surrogate errors, together with surrogate-induced misclassification and finite-sample estimation error bounds. Numerical experiments on multimodal benchmarks and PDE-driven rare-event problems up to 100 dimensions show that the proposed method achieves accuracy comparable to true-model adaptive importance sampling while requiring substantially fewer high-fidelity evaluations.

Data augmentation is widely recognized for improving generalization in deep networks, yet its impact on the geometry of learned representations remains poorly understood. In this work, we characterize how different data augmentation strategies reshape internal representations in neural networks. Using tools from shape analysis, we embed network hidden representations into a metric space where distance is invariant to scaling, translation, rotation and reflection. We show that increasing augmentation strength leads to well-behaved trajectories in this space, and that different augmentation types steer representations in distinct directions. Moreover, we investigate how neural representation shapes are distorted along data augmentation trajectories, and show that insights from neural geometry can predict which representations provide the most improvement when ensembling models. Our results reveal shared geometric patterns across architectures and seeds, and suggest that analyzing shape-space trajectories offers a principled tool for understanding and comparing data augmentation methods.

Interval-censored data arise frequently in scientific studies, where the event of interest is known only to occur within a specific time interval. In such studies, functional covariates taking the form of continuous curves or spatial profiles are increasingly encountered, and it is of substantial scientific relevance to investigate how the trajectory of a functional covariate affects the event time. We formulate the effects of both scalar and functional covariates on the interval-censored event time through a functional Cox model. We consider penalized maximum likelihood estimation for this model and devise an EM algorithm to stably compute the parameter estimators. The resulting estimators for the regression parameters and linear functionals of the coefficient function are shown to be consistent and asymptotically normal, with limiting covariance matrices that attain the semiparametric efficiency bound and can be readily estimated through the profile likelihood method. Building upon these results, we construct a global test for the overall effect of the functional covariate. Finally, we assess the performance of the proposed methods through extensive simulation studies and present an application to data from the Alzheimer's Disease Neuroimaging Initiative.

Spatial domain identification requires jointly modeling molecular signatures and physical coordinates, yet current tools frequently over-smooth biological boundaries, require user-specified cluster numbers, and lack principled multimodal integration. We introduce BaySC, an integrative Bayesian spatial clustering framework for spatial domain identification. BaySC inherently learns the true number of spatial domains from the data by employing a Mixture of Finite Mixtures (MFM) prior. Tissue topology is modeled via a Markov Random Field (MRF) applied to discrete cellular assignments, a strategy that enforces local spatial coherence without distorting the underlying gene expression features. This enables BaySC to accurately map contiguous tissue layers as well as geographically scattered, transcriptionally identical cell populations. Furthermore, BaySC handles spatial multi-omics data through a weighted log-likelihood fusion mechanism executed via Gibbs sampling. This approach assigns interpretable weights to each modality, allowing users to quantify the biological relevance of different data layers to the final tissue map. Validated across ten single-modal spatial transcriptomics and two spatial multi-omics datasets, BaySC yields highly interpretable probabilistic outputs. It demonstrates competitive accuracy on standard clustering metrics and consistently outperforms existing tools in preserving spatial topography, as measured by spatially-aware Adjusted Rand Index (spARI).

Operator-valued concentration inequalities are foundational to the analysis of modern high-dimensional statistics and randomized algorithms. However, standard oracle bounds are frequently limited in practice: they require explicit a priori knowledge of the true variance, and often explicitly scale with the ambient dimension, rendering them vacuous for infinite-dimensional or heavily structured operators. Motivated by these challenges, we establish the first empirical Bennett and Bernstein inequalities for sums of independent, bounded, compact self-adjoint operators. Our fully data-driven bounds replace the unknown variance with an empirical estimate and rely strictly on the intrinsic dimension rather than the ambient dimension. This structural shift yields computable, dimension-free guarantees that are strictly sharper for non-isotropic random matrices and seamlessly extend to infinite-dimensional Hilbert spaces. We demonstrate that our empirical bounds achieve asymptotic sharpness with the best known oracle rates. Finally, as an independent byproduct, we derive novel empirical concentration guarantees for the intrinsic dimension itself.

This paper investigates the critical role of eigenalignments between the kernel matrix and learning targets in achieving robust generalization in learning problems. We establish a direct connection between generalization performance in kernel methods and the estimation of eigenvectors and eigenvalues of matrices, offering a more intuitive understanding compared to prior work with minimal assumptions. We also show that, since the prediction task in KRR is essentially the weighted sum of eigenvectors/singular vectors, by analyzing how much error can be caused by perturbations to the kernel matrix, we can then derive a bound on this generalization error using the estimation stability of matrix eigenvalues and eigenvectors. Compared with previous work, our analysis concentrates on finite-sample settings and on the generalization error arising from having a suboptimal finite training set. Our findings reveal that in kernel methods, as long as the kernel is of high rank, the near-zero reconstruction error can be trivially obtained, implying that the reconstruction error will have limited predictive power for generalization. Finally, we establish a generalization bound from an eigenvalues/eigenvectors estimation perspective, showing that strong generalization requires increasing eigenvector alignment, eigenvalue magnitude, or gaps between consecutive eigenvalues.

Many complex systems can be reduced to their key components through spectrally decomposing matrices that capture their dynamics. These matrices can in turn be constructed from data, often by least-squares fitting: examples of algorithms to do this include Dynamical Mode Decomposition and variants, subspace identification and eigenvalue realisation algorithms. Typical outputs of these algorithms include a range of isolated, peripheral eigenvalues capturing persistent emergent patterns in the system. However, there is no objective way to assess which of these discrete eigenvalues are artefacts of finite data error, and which are reflections of a fully sampled operator. n this paper, we present a sampling pseudospectrum $P(λ)$, that provides probabilistic information on the behaviour of finite-data eigenvalues in the complex plane, and an estimator $\hat P(λ)$, which can be obtained by reprocessing our finite data sample. The estimator, which is computationally efficient to implement, allows us to test statistically for the location of the true eigenvalues. This gives us a rigorous and very general way to assess whether the patterns we extract from finite data are likely to be signal or noise.

By far the most common way to estimate an expected loss in machine learning is to draw samples, compute the loss on each one, and take the empirical average. However, sampling is not necessarily optimal. Given an MLP at initialization, we show how to estimate its expected output over Gaussian inputs without running samples through the network at all. Instead, we produce approximate representations of the distributions of activations at each layer, leveraging tools such as cumulants and Hermite expansions. We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling. We find moreover that our methods perform particularly well at estimating the probabilities of rare events, and additionally demonstrate how they can be used for model training. Together, these findings suggest a path to producing models with a greatly reduced probability of catastrophic tail risks.

Stochastic models of diffusion are routinely used to study dispersal of populations, including populations of animals, plants, seeds and cells. Advances in imaging and field measurement technologies mean that data are often collected across a range of scales, including count data collected across a series of fixed sampling regions to characterize population-level dispersal, as well as individual trajectory data to examine at the motion of individuals within a diffusive population. In this work we consider a lattice-based random walk model and examine the extent to which model parameters can be determined by collecting count data and/or trajectory data. Our analysis combines agent-based stochastic simulations, mean-field partial differential equation approximations, likelihood-based estimation, identifiability analysis, and model-based prediction. These combined tools reveal that working with count data alone can sometimes lead to challenges involving structural non-identifiability that can be alleviated by collecting trajectory data. Furthermore, these tools allow us to explore how different experimental designs impact inferential precision by comparing how different trajectory data collection protocols affects practical identifiability. Open source implementations of all algorithms used in this work are available on GitHub.

We propose a new probabilistic algorithm of $p$-adic linear regression for random sampling with digitwise noise. This includes a new probabilistic algorithm of modulo $p$ linear regression.

We present the first regret bound for classical online Q-learning in infinite-horizon discounted Markov decision processes (MDPs), without relying on optimism or bonus terms. We first analyze Boltzmann Q-learning with decaying temperature and show that its regret depends critically on the suboptimality gap of the MDP: for sufficiently large gaps, the regret is sublinear, while for small gaps it deteriorates and can approach linear growth. To address this limitation, we study a Smoothed $ε_n$-Greedy exploration scheme that combines $ε_n$-greedy and Boltzmann exploration, for which we prove a gap-robust regret bound of near-$\tilde{O}(N^{9/10})$. We also obtain sample complexity guarantees, with both regret and sample complexity bounds holding with high probability. To analyze these algorithms, we develop a high-probability concentration bound for contractive Markovian stochastic approximation with iterate- and time-dependent transition dynamics. This bound may be of independent interest as the contraction factor in our framework is allowed to converge to one asymptotically.

We show that high-accuracy guarantees for log-concave sampling -- that is, iteration and query complexities which scale as $\mathrm{poly}\log(1/δ)$, where $δ$ is the desired target accuracy -- are achievable using stochastic gradients with subexponential tails. Notably, this exhibits a separation with the problem of convex optimization, where stochasticity (even additive Gaussian noise) in the gradient oracle incurs $\mathrm{poly}(1/δ)$ queries. We also give an information-theoretic argument that light-tailed stochastic gradients are necessary for high accuracy: for example, in the bounded variance case, we show that the minimax-optimal query complexity scales as $Θ(1/δ)$. Our framework also provides similar high accuracy guarantees under stochastic zeroth order (value) queries, and an improved complexity result for sampling from finite-sum potentials.

Inference in nonlinear continuous stochastic processes on trees is challenging, particularly when observations are sparse and the topology is complex. Exact smoothing via Doob's $h$-transform is intractable for general nonlinear dynamics. We propose Neural Backward Filtering Forward Guiding (NBFFG), a unified framework for both discrete transitions and continuous diffusions. Our method constructs a variational posterior by leveraging a proxy linear-Gaussian process. This proxy process yields a closed-form backward filter that serves as a guide, steering the generative path toward high-likelihood regions. We then learn a neural residual to capture the non-linear discrepancies. This formulation allows for an unbiased pathwise subsampling scheme, reducing the training complexity from tree-size dependent to path-length dependent. Empirical results show that NBFFG outperforms baselines on synthetic benchmarks, and we demonstrate the method on a high-dimensional inference task in phylogenetic analysis with reconstruction of ancestral butterfly wing shapes.

Partial Least Squares (PLS) learns shared structure from paired data via the top singular vectors of the empirical cross-covariance (PLS-SVD), but multimodal datasets often have missing entries in both views. We study PLS-SVD under independent entry-wise missing-completely-at-random masking in a proportional high-dimensional spiked model. After appropriate normalization, the masked cross-covariance behaves like a spiked rectangular random matrix whose effective signal strength is attenuated by $\sqrtρ$, where $ρ$ is the joint entry retention probability. The replica-symmetric analysis predicts a sharp BBP-type phase transition: below a critical signal-to-noise threshold the leading singular vectors are asymptotically uninformative, while above it they achieve nontrivial alignment with the latent shared directions, with closed-form asymptotic overlap formulas. We also state a finite-rank extension as a conjecture, predicting that the same missingness-adjusted threshold applies componentwise when the latent spikes are separated. Simulations and semi-synthetic multimodal experiments agree with the predicted phase diagram and recovery curves across aspect ratios, signal strengths, and missingness levels.

Non-negative matrix factorization (NMF) approximates a non-negative endogenous data matrix as $Y_1 \approx XB$, with non-negative latent components $X$ and coefficients $B$. Standard covariate-aware NMF is feedforward: $B$ depends only on exogenous variables $Y_2$, with no latent feedback among endogenous variables. We propose NMF-FFB (NMF with feedforward-feedback structure), an exploratory data-fitting framework that embeds the simultaneous equation $B = Θ_1 Y_1 + Θ_2 Y_2$ in NMF, where $Θ_1$ is non-negative latent feedback and $Θ_2$ non-negative exogenous pathways. NMF-FFB is positioned within data-fitting structural equation modeling (SEM): it fits $Y_1$ directly rather than a model-implied covariance, and is not a confirmatory measurement model or a replacement for maximum-likelihood SEM under standard confirmatory factor analysis assumptions. When $ρ(XΘ_1)<1$, the reduced form $Y_1 \approx (I-XΘ_1)^{-1} XΘ_2 Y_2$ defines a latent Leontief inverse separating direct from cumulative feedback-amplified effects. Estimation uses regularized multiplicative updates with orthogonality and sparsity penalties; an $X$-fixed bootstrap summarizes uncertainty for the feedback spectral radius, the amplification ratio, and path coefficients. Unlike conventional SEM, NMF-FFB requires only the latent rank $Q$ and lets $X$ group endogenous indicators into latent factors. This suits non-negative additive data, automatic loading discovery, Leontief-type cumulative effects, and small samples where covariance-based maximum-likelihood fitting is ill-conditioned. Applications to Holzinger-Swineford, Los Angeles pollution-mortality, and Mississippi county-level health data demonstrate interpretable parts-based representations across distinct latent-feedback regimes.

Equivariant neural networks encode the intrinsic symmetry of data as an inductive bias, which has achieved impressive performance in wide domains. However, the understanding to their expressive power remains premature. Focusing on 2-layer ReLU networks, this paper investigates the impact of enforcing equivariance constraints on the expressive power. By examining the boundary hyperplanes and the channel vectors, we constructively demonstrate that enforcing equivariance constraints could undermine the expressive power. Naturally, this drawback can be compensated for by enlarging the model size -- we further prove upper bounds on the required enlargement for compensation. Surprisingly, we show that the enlarged neural architectures have reduced hypothesis space dimensionality, implying even better generalizability.

Sheaf Neural Networks equip graph structures with a cellular sheaf: a geometric structure which assigns local vector spaces (stalks) and a linear learnable restriction/transport maps to nodes and edges, yielding an edge-aware inductive bias that handles heterophily and limits oversmoothing. However, common Neural Sheaf Diffusion implementations rely on SVD-based sheaf normalization and dense per-edge restriction maps, which scale with stalk dimension, require frequent Laplacian rebuilds, and yield brittle gradients. To address these limitations, we introduce Polynomial Neural Sheaf Diffusion (PolyNSD), a new sheaf diffusion approach whose propagation operator is a degree-K polynomial in a normalised sheaf Laplacian, evaluated via a stable three-term recurrence on a spectrally rescaled operator. This provides an explicit K-hop receptive field in a single layer (independently of the stalk dimension), with a trainable spectral response obtained as a convex mixture of K+1 orthogonal polynomial basis responses. PolyNSD enforces stability via convex mixtures, spectral rescaling, and residual/gated paths, reaching new state-of-the-art results on both homophilic and heterophilic benchmarks, inverting the Neural Sheaf Diffusion trend by obtaining these results with just diagonal restriction maps, decoupling performance from large stalk dimension, while reducing runtime and memory requirements.

Self-supervised learning (SSL) has recently advanced through non-contrastive methods that couple an invariance term with variance, covariance, or redundancy-reduction penalties. While such objectives shape first- and second-order statistics of the representation, they largely ignore the local geometry of the underlying data manifold. In this paper, we introduce CurvSSL, a curvature-regularized self-supervised learning framework, and its RKHS extension, kernel CurvSSL. Our approach retains a standard two-view encoder-projector architecture with a Barlow Twins-style redundancy-reduction loss on projected features, but augments it with a curvature-based regularizer. Each embedding is treated as a vertex whose $k$ nearest neighbors define a discrete curvature score via cosine interactions on the unit hypersphere; in the kernel variant, curvature is computed from a normalized local Gram matrix in an RKHS. These scores are aligned and decorrelated across augmentations by a Barlow-style loss on a curvature-derived matrix, encouraging both view invariance and consistency of local manifold bending. Experiments on MNIST and CIFAR-10 datasets with a ResNet-18 backbone show that curvature-regularized SSL yields competitive or improved linear evaluation performance compared to Barlow Twins and VICReg. Our results indicate that explicitly shaping local geometry is a simple and effective complement to purely statistical SSL regularizers.

The study of self-normalized processes plays a crucial role in a wide range of applications, from sequential decision-making to econometrics. While the behavior of self-normalized concentration has been widely investigated for scalar-valued processes, vector-valued processes remain comparatively underexplored, especially outside of the sub-Gaussian framework. In this contribution, we provide concentration bounds for self-normalized processes with light tails beyond sub-Gaussianity (such as Bennett or Bernstein bounds). We illustrate the relevance of our results in the context of online linear regression, with applications in (kernelized) linear bandits.

Hybrid type 2 studies are gaining popularity for their ability to assess both implementation and health outcomes as co-primary endpoints. Often conducted as cluster-randomized trials (CRTs), five design methods can validly power these studies: p-value adjustment methods, combined outcomes approach, single weighted 1-DF test, disjunctive 2-DF test, and conjunctive test. We compared these methods theoretically and numerically. Theoretical comparisons of power equations allowed us to identify when one method had more or less power than another globally. We showed that p-value adjustment methods are always less powerful than both the combined outcomes approach and the single 1-DF test, and identified conditions where the disjunctive 2-DF test is less powerful than the single 1-DF test. To further investigate when power advantages shift, we conducted a large-scale numerical study using our novel crt2power R package, which calculates power or sample size for CRTs with two continuous co-primary endpoints using these methods. Across 45,000 input scenarios, we found specific patterns: when treatment effects are unequal, the disjunctive 2-DF test tends to be most powerful; when treatment effects are equal, the single 1-DF test tends to dominate. Together, these comparisons offer practical guidance for powering hybrid type 2 studies.

We consider sampling from a Gibbs distribution by evolving finitely many particles. We propose a preconditioned version of a recently proposed noise-free sampling method, governed by approximating the score function with the numerically tractable score of a regularized Wasserstein proximal operator. This is derived by a Cole--Hopf transformation on coupled anisotropic heat equations, yielding a kernel formulation for the preconditioned regularized Wasserstein proximal. The diffusion component of the proposed method is also interpreted as a modified self-attention block, as in transformer architectures. For quadratic potentials, we provide a discrete-time non-asymptotic convergence analysis and explicitly characterize the bias, which is dependent on regularization and independent of step-size. Experiments demonstrate acceleration and particle-level stability on various log-concave and non-log-concave toy examples to Bayesian total-variation regularized image deconvolution, and competitive/better performance on non-convex Bayesian neural network training when utilizing variable preconditioning matrices.

We conclude our work [arXiv:2403.07628, arXiv:2503.12644] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the correction terms in the asymptotic expansion are multilinear forms of the higher-order derivatives of the leading-order term, with certain rational polynomial coefficients that are independent of the dummy generating function variable. In this way, the same multilinear structure, with the same polynomial coefficients, is inherited by the asymptotic expansion of any linearly induced quantity such as the distribution of the $k$-th largest level. Whereas the results for the unitary ensembles are presented with proof, the discussion of the orthogonal and symplectic ones is based on some hypotheses. To substantiate the hypotheses, we check the result for the $k$-th largest level in the orthogonal ensembles against simulation data for choices of $n$ and $k$ that require as many as four correction terms to achieve satisfactory accuracy.

In Generalised Bayesian Inference (GBI), the learning rate and hyperparameters of the loss must be estimated. These inference-hyperparameters can't be estimated jointly with the other parameters, from the data, by giving them a prior. However, in some settings there exist unknown ``true'' hyperparameter-values about which it is meaningful to have prior belief. It is then possible to use Bayesian inference with held-out data to get hyperparameter-posteriors. We define two hyperparameter posteriors, one based on an ELPPD-utility and one aiming to cover the pseudo-true parameter. The new framework supports estimation and uncertainty quantification for multiple hyperparameters jointly. Experiments show that the resulting GBI-posteriors out-perform Bayesian inference on simulated test data and select optimal or near optimal hyperparameter values in a large real problem of text analysis. Generalised Bayesian inference is particularly useful for combining multiple data sets and most of our examples belong to that setting. We also give asymptotic results for some of the special ``multi-modular'' Generalised Bayes posteriors which we use in our examples.

Will further scaling up of machine learning models continue to bring success? A significant challenge in answering this question lies in understanding generalization gap, which is the impact of overfitting. Understanding generalization gap behavior of increasingly large-scale machine learning models remains a significant area of investigation, as conventional analyses often link error bounds to model complexity, failing to fully explain the success of extremely large architectures. This research introduces a novel perspective by establishing a model-independent upper bound for generalization gap applicable to algorithms whose outputs are determined solely by the data's histogram, such as empirical risk minimization or gradient-based methods. Crucially, this bound is shown to depend only on the Rényi entropy of the data-generating distribution, suggesting that a small generalization gap can be maintained even with arbitrarily large models, provided the data quantity is sufficient relative to this entropy. This framework offers a direct explanation for the phenomenon where generalization performance degrades significantly upon injecting random noise into data, where the performance degrade is attributed to the consequent increase in the data distribution's Rényi entropy. Furthermore, we adapt the no-free-lunch theorem to be data-distribution-dependent, demonstrating that an amount of data corresponding to the Rényi entropy is indeed essential for successful learning, thereby highlighting the tightness of our proposed generalization bound.

A key trait of stochastic optimizers is that multiple runs of the same optimizer in attempting to solve the same problem can produce different results. As a result, their performance is evaluated over several repeats, or runs, on the problem. However, the accuracy of the estimated performance metrics depends on the number of runs and should be studied using statistical tools. We present a statistical analysis of the common metrics, and develop guidelines for experiment design to measure the optimizer's performance using these metrics to a high level of confidence and accuracy. To this end, we first discuss the confidence interval of the metrics and how they are related to the number of runs of an experiment. We then derive a lower bound on the number of repeats in order to guarantee achieving a given accuracy in the metrics. Using this bound, we propose an algorithm to adaptively adjust the number of repeats needed to ensure the accuracy of the evaluated metric. Our simulation results demonstrate the utility of our analysis and how it allows us to conduct reliable benchmarking as well as hyperparameter tuning and prevent us from drawing premature conclusions regarding the performance of stochastic optimizers.

This paper develops a performant Bayesian approach to conditional average treatment effect (CATE) estimation in regression discontinuity designs (RDD), an increasingly prevalent form of quasi-experiment that facilitates causal inference. Earlier Bayesian approaches do not easily accommodate CATE estimation while recent frequentist approaches to this problem assume a known basis expansion, a steep model specification requirement that our approach avoids. The new model is a variant of a Bayesian additive regression tree (BART) model with linear leaf-level regressions on the running variable and a treatment dummy (and their interaction). The model adaptively partitions covariate space into regions where the slope on the running variable appreciably differs, providing interpretable Bayesian inference on conditional average treatment effects near the cutoff.

Hallucinations, a phenomenon where a language model (LM) generates nonfactual content, pose a significant challenge to the practical deployment of LMs. While many empirical methods have been proposed to mitigate hallucinations, recent studies established a computability-theoretic result showing that any LM will inevitably generate hallucinations on an infinite set of inputs, regardless of the quality and quantity of training datasets and the choice of the language model architecture and training and inference algorithms. Although the computability-theoretic result may seem pessimistic, its significance in practical viewpoints has remained unclear. This paper claims that those "innate" inevitability results from computability theory and diagonal argument, in principle, cannot explain practical issues of LLMs. We demonstrate this claim by presenting a positive theoretical result from a probabilistic perspective. Specifically, we prove that hallucinations can be made statistically negligible, provided that the quality and quantity of the training data are sufficient. Interestingly, our positive result coexists with the computability-theoretic result, implying that while hallucinations on an infinite set of inputs cannot be entirely eliminated, their probability can always be reduced by improving algorithms and training data. By evaluating the two seemingly contradictory results through the lens of information theory, we argue that our probability-theoretic positive result better reflects practical considerations than the computability-theoretic negative result.

We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. Subsequently, such differences are integrated, weighted by their associated uncertainties, using a maximum-likelihood formulation. This procedure can be seen as an extension to a multidimensional setting of the thermodynamic integration, a technique developed in statistical physics. The method leverages the manifold hypothesis, estimating quantities within the intrinsic data manifold without defining an explicit coordinate map. It does not rely on any binning or space partitioning, but rather on the construction of a neighbourhood graph based on an adaptive bandwidth selection procedure. BMTI mitigates the limitations commonly associated with traditional nonparametric density estimators, effectively reconstructing smooth profiles even in high-dimensional embedding spaces. The method is tested on a variety of complex synthetic high-dimensional datasets, where it is shown to outperform traditional estimators, and is benchmarked on realistic datasets from the chemical physics literature.

In bipartite causal inference with interference, interventional units might receive treatment or control, and they might affect the outcome of outcome units through their connections on a bipartite network. We study bipartite causal inference with interference based on observational data across time and a changing bipartite network. Under an exposure mapping framework, we define the immediate and carryover causal effects for each outcome unit, representing contrasts of potential outcomes under different values of the immediately preceding and past exposures, respectively, averaged over time. We establish unconfoundedness of the exposure received by outcome units based on unconfoundedness assumptions on the interventional units' treatment assignment and the random network, hence respecting the bipartite structure of the problem. Our results hold for binary, continuous, and multivariate exposure mappings. In the special case of binary exposure and carryover mappings, we propose algorithms for the immediate and carryover causal effects that combine matching and covariate balancing. We show that the bias of the resulting estimators is bounded. In our motivating study, we find some evidence that smoke from wildfires has an immediate impact on reducing transportation by bicycle in San Francisco.

Operator learning is a rising field of scientific computing where inputs or outputs of a machine learning model are functions defined in infinite-dimensional spaces. In this paper, we introduce NEON (Neural Epistemic Operator Networks), an architecture for generating predictions with uncertainty using a single operator network backbone, which presents orders of magnitude less trainable parameters than deep ensembles of comparable performance. We showcase the utility of this method for sequential decision-making by examining the problem of composite Bayesian Optimization (BO), where we aim to optimize a function $f=g\circ h$, where $h:X\to C(\mathcal{Y},\mathbb{R}^{d_s})$ is an unknown map which outputs elements of a function space, and $g: C(\mathcal{Y},\mathbb{R}^{d_s})\to \mathbb{R}$ is a known and cheap-to-compute functional. By comparing our approach to other state-of-the-art methods on toy and real world scenarios, we demonstrate that NEON achieves state-of-the-art performance while requiring orders of magnitude less trainable parameters.

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

Despite advances in representation learning, high-dimensional classification remains challenging in low-sample-size regimes, where the dominant signal may vary across applications and labeled data are often limited. We propose a dissimilarity-profiling classification framework that represents each observation by its class-wise dissimilarity profile, transforming the original feature space into a low-dimensional representation that summarizes how the observation relates to each class. The key idea is to turn a consequence of the curse of dimensionality into signal: high-dimensional geometry can induce systematic within-class and between-class dissimilarity patterns under location, scale, or other distributional changes, and these patterns are captured by the class-wise profiles. Building on this representation, we introduce a rank-transformed algorithm that converts dissimilarities into class-wise rank profiles, yielding a compact representation for classification. The proposed method delivers competitive or improved performance relative to commonly used classifiers on two-class, multi-class, network, and real high-dimensional low-sample-size datasets. To provide insight into the mechanism underlying the method, we analyze a distance-based surrogate and show that the resulting profiles encode differences in first, second, and higher-order moments, while the rank transformation improves robustness to outliers. Together, these results show that rank-transformed dissimilarity profiles provide an adaptive representation for high-dimensional classification when the signal structure is unknown.

In the eleventh and twelfth centuries in England, Wales and Normandy, Royal Acta were legal documents in which witnesses were listed in order of social status. Any bishops present were listed as a group. For our purposes, each witness-list is an ordered permutation of bishop names with a known date or date-range. Changes over time in the order bishops are listed may reflect changes in their authority. Historians would like to detect and quantify these changes. There is no reason to assume that the underlying social order which constrains bishop-order within lists is a complete order. We therefore model the evolving social order as an evolving partial ordered set or {\it poset}. We construct a Hidden Markov Model for these data. The hidden state is an evolving poset (the evolving social hierarchy) and the emitted data are random total orders (dated lists) respecting the poset present at the time the order was observed. This generalises existing models for rank-order data such as Mallows and Plackett-Luce. We account for noise via a random ``queue-jumping'' process. Our latent-variable prior for the random process of posets is marginally consistent. A parameter controls poset depth and actor-covariates inform the position of actors in the hierarchy. We fit the model, estimate posets and find evidence for changes in status over time. We interpret our results in terms of court politics. Simpler models, based on Bucket Orders and vertex-series-parallel orders, are rejected. We compare our results with a time-series extension of the Plackett-Luce model. Our software is publicly available.

Bayesian statistical inference loses predictive optimality when generative models are misspecified. Working within an existing coherent loss-based generalisation of Bayesian inference, we show existing Modular/Cut-model inference is coherent, and write down a new family of Semi-Modular Inference (SMI) schemes, indexed by an influence parameter, with Bayesian inference and Cut-models as special cases. We give a meta-learning criterion and estimation procedure to choose the inference scheme. This returns Bayesian inference when there is no misspecification. The framework applies naturally to Multi-modular models. Cut-model inference allows directed information flow from well-specified modules to misspecified modules, but not vice versa. An existing alternative power posterior method gives tunable but undirected control of information flow, improving prediction in some settings. In contrast, SMI allows tunable and directed information flow between modules. We illustrate our methods on two standard test cases from the literature and a motivating archaeological data set.

Multiple imputation is a highly recommended technique to deal with missing data, but the application to longitudinal datasets can be done in multiple ways. When a new wave of longitudinal data arrives, we can treat the combined data of multiple waves as a new missing data problem and overwrite existing imputations with new values (re-imputation). Alternatively, we may keep the existing imputations, and impute only the new data. We may do either a full multiple imputation (nested) or a single imputation (appended) on the new data per imputed set. This study compares these three strategies by means of simulation. All techniques resulted in valid inference under a monotone missingness pattern. A non-monotone missingness pattern led to biased and non-confidence valid regression coefficients after nested and appended imputation, depending on the correlation structure of the data. Correlations within timepoints must be stronger than correlations between timepoints to obtain valid inference. In an empirical example, the three strategies performed similarly.We conclude that appended imputation is especially beneficial in longitudinal datasets that suffer from dropout.

We present and compare multiple imputation methods for multilevel continuous and binary data where variables are systematically and sporadically missing. The methods are compared from a theoretical point of view and through an extensive simulation study motivated by a real dataset comprising multiple studies. Simulations are reproducible. The comparisons show why these multiple imputation methods are the most appropriate to handle missing values in a multilevel setting and why their relative performances can vary according to the missing data pattern, the multilevel structure and the type of missing variables. This study shows that valid inferences can only be obtained if the dataset gathers a large number of clusters. In addition, it highlights that heteroscedastic MI methods provide more accurate inferences than homoscedastic methods, which should be reserved for data with few individuals per cluster. Finally, the method of Quartagno and Carpenter (2016a) appears generally accurate for binary variables, the method of Resche-Rigon and White (2016) with large clusters, and the approach of Jolani et al. (2015) with small clusters.