The proliferation of generative artificial intelligence has given rise to an interactive learning environment, where model parameters are continuously updated using not only data generated by natural processes, but also synthetic outputs produced by other models. This paradigm introduces two major challenges: (1) training data are no longer drawn exclusively from the target population, undermining a core assumption of classical statistical learning, and (2) model training processes become inherently correlated, as models interact with one another through repeated exposure to each other's synthetic outputs in a potentially complex manner. Establishing reliable statistical inference in such structured interactive learning environments therefore remains an important open problem. In particular, there is growing concern about model collapse, a phenomenon in which the performance of generative models progressively degrades as they are trained on synthetic data produced by earlier model generations. Prior work on model collapse primarily focuses on a single model trained on its own output, failing to capture model performance in multi-model interactive settings. In this work, we fill this gap by investigating the performance of generative models in an interactive learning environment with general interaction patterns. In particular, we formalize model interactions using directed graphs and show that the occurrence of model collapse depends critically on the topology of the interaction graph. We further derive an explicit necessary and sufficient condition characterizing when model collapse occurs, and establish finite-sample results for linear regression and asymptotic guarantees for general M-estimators. We support our theoretical findings through extensive numerical experiments.

Bayesian optimization (BO) selects evaluation points for expensive black-box objectives using Gaussian process (GP) predictive distributions. Kernel choice and hyperparameter selection can lead to miscalibrated predictive distributions and an inappropriate exploration-exploitation trade-off. For minimization, sampling criteria such as expected improvement (EI) depend on the predictive distribution below the current best value, so lower-tail miscalibration directly affects the sampling decision. This article studies goal-oriented calibration of GP predictive distributions below a low threshold $t$ in the noiseless setting, for standard GP models with hyperparameters selected by maximum likelihood. A framework for predictive reliability below $t$ is introduced, based on two notions of spatial calibration: occurrence calibration over the design space and thresholded $μ$-calibration on sublevel sets of the form $\{x\in\mathbb{X}, f(x)\le t\}$. Building on this framework, we propose tcGP, a post-hoc method that calibrates GP predictive distributions below~$t$, and we show that the resulting EI-based global optimization algorithm remains dense in the design space. Experiments on standard benchmarks show improved lower-tail calibration and BO performance relative to standard GP models and globally calibrated GP models.

Risk management is an important part of financial practice, essential for protecting assets and investments in modern-day volatile markets. This paper proposes a mixture of mirrored Weibull (MMW) distribution for modelling stock returns and estimating risk measures. Unlike common practices which are typically based on the normal distribution, the MMW model can flexibly accommodate non-normal features frequently exhibited in financial data. It also enjoys appealing properties such as having a simple density expression and fast parameter estimation. We demonstrate the effectiveness of our model by assessing its performance in Value-at-Risk (VaR) estimation of three S&P500 stocks. The MMW model compares favourably to Gaussian mixture model and t-mixture model, with significant improvements in VaR estimation and prediction.

In clinical studies, persistence, which measures the duration of time a patient continues to take a prescribed medication without discontinuation, is increasingly recognized as a critical indicator of adherence to medication. Adherence encompasses not only whether a patient takes their medication as prescribed but also the consistency and duration with which they do so. Among the various metrics used to evaluate adherence, persistence stands out as a particularly robust measure because it provides a temporal dimension, reflecting the sustained commitment of patients to their therapeutic regimens. This focus on persistence offers unique insights into adherence-related quality and performance, shedding light on the challenges and opportunities to optimize long-term medication use. The comparison of upper-tail clinical performance, which measures the extent to which very large responses persist among top responders, is often more decisive in therapy evaluation than conventional summaries. In this paper, we introduce the quantile-based effectiveness persistence function defined as the ratio between the tail mean and the quantile function. The notion parallels expected shortfall in risk theory and is tailored to detect clinically meaningful deviations in the upper tail. We establish key properties and show that the function is equivalent to the first L-moment of the scaled tail, yielding robust inference tools. We derive a simple nonparametric estimator of the function and develop a bootstrap-calibrated two-sample (upper-tail) equivalence test. Simulation studies and real-data analysis illustrate that the proposed measures captures clinically relevant tail persistence that complements median and mean-based summaries.

Privacy constraints have driven the rise of federated learning (FL), which enables multi-site analyses without sharing individual participant data. We develop a framework for FL with missing data, identifying conditions under which the complete case (CC) estimator is preferred over the inverse probability weighting (IPW) estimator. For settings where the CC estimator fails, we introduce a calibrated weight estimation approach that combines candidate weighting models across sites and remains consistent if at least one is correctly specified. Consistency conditions are stated at the site level, ensuring that the federated estimator inherits validity from local properties. We derive a sandwich variance estimator that accounts for uncertainty in weight estimation, and illustrate the framework by evaluating risk factors for 90-day mortality among patients with pleural infections treated with intrapleural enzyme therapy.

Squared Wasserstein distance is a frequently used tool to measure discrepancy between probability distributions. This distance is typically computed between empirical measures of size $n$ from two underlying random samples. Unfortunately, even in lower dimensional Euclidean space problems $\left( d \in \{2,3\} \right)$, algorithms for Wasserstein distance computation with approximate or exact precision guarantees scale poorly in the runtime as a function of $n$ and the desired precision. In response, we consider the computational-statistical runtime, where the goal is to estimate from samples the Wasserstein distance between potentially smooth measures up to $ε$-additive error in expectation with respect to the sampling; we allow $O(1)$ computational cost for collecting a sample. Towards this, we develop a Sample-Sketch-Solve paradigm where we introduce a regular cartesian grid sketch of the samples. We show that (especially under $α$-Hölder smooth distributions) this can compress the data without increasing asymptotic error, and also regularizes the structure which enables faster exact algorithms. Ultimately, we approximate $W_2^2(P,Q)$ within $ε$ error in $ε^{-\max(2,\frac{d+1+o(1)}{1+α})}$ time for $0 < α< 1$ Hölder smooth distributions $P,Q$ on $(0,1)^{d}$; an optimal $Θ(ε^{-2})$ for $α> 1/2$ when $d=2$ and nearly optimal as $α\to 1$ when $d = 3$.

Independent component (IC) models are a standard tool for representing multivariate data in statistics, signal processing, and machine learning. Despite the extensive use of IC models, much less attention has been given to goodness-of-fit tests for assessing their compatibility with data. We develop the first goodness-of-fit test for IC models that is supported by a theoretical validity guarantee when the data dimension and sample size diverge proportionally. This is made possible by the fact that the test does not rely on a pre-whitening step, which often limits the applicability of other goodness-of-fit tests in high dimensions. Our theoretical analysis is complemented with numerical experiments that demonstrate the test's size control and power under a range of conditions. In addition, we provide examples involving gene-expression data to illustrate that the test has potential for effective diagnostic use in practice.

Standard generative models struggle with heavy-tailed data: Lipschitz architectures cannot produce power-law tails from Gaussian noise, and interpolating between heavy-tailed data and Gaussians is ill-posed. We propose a simple fix: apply the soft-log transform $ϕ(x) = \mathrm{sign}(x) \cdot \log(1 + |x|)$ coordinate-wise to data before training, then exponentiate samples after generation. A Hill diagnostic decides per-coordinate whether to transform, leaving light-tailed margins untouched at no added complexity. This compresses heavy tails into a range where standard flow matching succeeds, without heavy-tailed base distributions or architectural modifications. We provide theoretical intuition for why this works: the log-transform maps Pareto tails to exponentials, and the induced dynamics implement a form of tail annealing via power transformations. On a 144-configuration multivariate benchmark (3 copulas, $d$ up to 100, 4 tail indices), Log-FM dominates specialized baselines on $W_1$, CVaR$_{99}$, and extreme-quantile metrics, and is the only method with zero severe divergences across 2{,}880 runs.

Existing identification results in proximal causal inference often focus on marginal interventional distributions using standard outcome or treatment bridge functions. These methods do not generally identify joint interventional distributions that contain all proxy variables that were used to define the corresponding bridge functions. In many applications, however, these joint interventional distributions are a natural target of interest. We introduce extended bridge functions and derive new identification results for joint interventional distributions that may retain all relevant proxy variables. We then apply these results to proximal identification algorithms, where interventional kernels naturally arise as intermediate objects, yielding a generalized framework based on kernel operations.

In this work, we study the estimation of treatment duration effects in observational survival data, where treatment and covariate histories evolve over time and longer observed durations are only attainable among individuals who remain event-free and under follow-up, leading to immortal time bias under naive analyses. The cloning-censoring-weighting (CCW) framework provides a practical approach to emulate target trials of treatment duration strategies, but several methodological aspects remain insufficiently understood. We focus on static treatment duration strategies under two settings of increasing complexity: baseline confounding only, and confounding with time-varying covariates. We formalize the assumptions underlying CCW, with particular emphasis on treatment admissibility, relaxed intervention rules, and the distinction between artificial and natural censoring. We then compare several estimation approaches after cloning and censoring, including inverse probability of censoring weighting (IPCW), the G-formula, and doubly robust estimators, through simulation studies assessing robustness, variability, and sensitivity to censoring model misspecification. Finally, we apply the framework to a Breast Cancer cohort to emulate a target trial comparing 2 versus 5 years of adjuvant tamoxifen in early stage breast cancer. Due to the small number of events and limited support for the 2-year strategy, estimates are associated with substantial uncertainty. These findings highlight both the practical relevance and the limitations of CCW, and underscore the importance of sensitivity analyses in complex longitudinal observational settings.

Importance sampling with data-driven proposal distributions is widely used in practice. A common workflow first generates an auxiliary sample of size $N$ from an approximation of the target distribution, constructs a density estimate $\hat q$ such as a kernel density estimator (KDE), and then draws $n$ importance samples from this learned proposal. Despite its practical relevance, the theoretical properties of this hierarchical procedure remain poorly understood, since classical importance sampling theory assumes a fixed proposal. We address this gap by deriving non-asymptotic error bounds for standard, defensive, and self-normalized importance sampling estimators with random proposals. Our results separate the Monte Carlo error, scaling as $n^{-1/2}$, from the proposal approximation error measured through the mean integrated absolute and squared errors (MIAE and MISE) of $\hat q$. To obtain explicit convergence rates in $(N,n)$, we establish MIAE and MISE bounds for KDEs constructed from geometrically ergodic Markov chains in stationary and non-stationary regimes. Combining these results yields quantitative guarantees for importance sampling with KDE-based proposals. Our theory provides practical guidance for selecting defensive mixture weights in a nonparametric importance sampling framework.

This paper studies a uniform projection criterion for space-filling designs under the stratified $L_2$-discrepancy. The criterion, denoted by $Φ_{SD}$, is the average squared stratified $L_2$-discrepancy over all two-dimensional projections. For U-type $(n,m,s^p)$ designs, we derive an explicit formula for $Φ_{SD}$ in terms of row-pairwise weighted hierarchical distances, and we establish sharp lower and upper bounds with equality conditions. We further show that many known optimal constructions attain the lower bound of $Φ_{SD}$, and that designs attaining the lower bound of the full stratified $L_2$-discrepancy also attain the lower bound of $Φ_{SD}$. The criterion can be evaluated in $O(n^2m)$ time, with a modest reduction in arithmetic operations compared with direct projection-wise evaluation. Numerical studies illustrate the theoretical results and show that $Φ_{SD}$ is effective for assessing low-dimensional projection uniformity.

Motivated by multimodal estimation, we study a multi-view spiked Wigner model in which several noisy matrix observations contain correlated latent spikes. We derive a spectral estimator for the latent spikes by linearizing approximate message passing (AMP). Our main result is an explicit sharp transition formula for its spectrum: for $L \geq 2$ views, letting $λ$ be the $L$-dimensional vector of spike strengths and $B$ the $L\times L$ limiting Gram matrix of the spikes, the critical parameter is $\mathsf{SNR}(λ,B)=λ_{\max}[\mathrm{Diag}(\sqrtλ) (B \odot B) \mathrm{Diag}(\sqrtλ)]$. When $\mathsf{SNR}(λ,B)<1$, the linearized AMP matrix has no outlier beyond the right edge of its bulk spectrum. When $\mathsf{SNR}(λ,B)>1$, an informative outlier is pinned at the distinguished point $1$, and the associated eigenvector has explicit, nontrivial overlaps with the latent signals. Thus $\mathsf{SNR}(λ,B)=1$ gives the exact spectral weak-recovery threshold for the linearized AMP method. To establish our results, we analyze the correlated Gaussian noise matrix through a matrix Dyson equation and combine this deterministic description with finite-rank perturbation arguments adapted to the multi-view spike structure. We also show that, for a broad class of spike priors, the spectral threshold $\mathsf{SNR}(λ,B)=1$ coincides with the information-theoretic threshold for weak recovery, ruling out a statistical-computational gap for this class of priors.

Tolerance limits have received considerable attention in the statistical literature, with applications reaching far beyond their initial role in quality control. The well-known formula of Scheffé and Tukey (1944) establishes a simple, distribution-free relation between sample size and population coverage by two given order statistics and a given confidence level. A key requirement in applying this formula is the availability of an unbiased, representative sample from the population of interest. However, as it often happens in biological and medical applications, various logistical constraints may preclude the possibility of obtaining an unbiased sample. We derive extensions of this formula which accommodate a large class of biased sampling schemes including weight bias and censoring. The modified formulae are validated through a simulation study and compared to its unmodified counterpart. We illustrate the use of the modified formulae using the partially observed failure times for individuals with dementia using data collected from the Canadian Study of Health and Aging.

Stationary subspace analysis (SSA) is a blind source separation framework that decomposes linearly mixed multivariate data into stationary and nonstationary components. We extend SSA to spatially indexed data by introducing spatial stationary subspace analysis (spSSA), which explicitly accounts for spatial dependence. We propose three estimation procedures for the unmixing matrix based on first- and second-order spatial statistics. Each procedure targets a different type of nonstationarity and can be formulated as the solution to a generalized eigenvalue problem. To address situations where multiple forms of nonstationarity are present simultaneously, we combine the three procedures using approximate joint diagonalization. Simulation studies demonstrate that this combined approach yields superior separation performance. When the dimension of the nonstationary subspace is known, the proposed methods reliably recover the latent stationary and nonstationary components. However, determining this dimension remains a fundamental challenge in SSA, for which no generally accepted solution currently exists. Building on our estimation procedures, we propose a novel data augmentation approach to estimate the dimension of the nonstationary subspace and demonstrate its effectiveness through simulation studies. The proposed methodology is easily transferable to time series settings, making it of broader methodological interest.

Conventional treatment policies map patient covariates to a single recommended intervention in order to maximize expected clinical outcomes. Although a rich body of causal inference methods has been developed to estimate such policies, point-valued recommendations can be highly sensitive to estimation uncertainty, model specification, and finite-sample variability, while typically providing little guidance about how confident one should be in the recommended action. In this work, we propose a set-valued policy learning paradigm for the multiple-treatment setting, in which policies output a set of plausible treatments rather than a single recommendation. This formulation enables intrinsic uncertainty quantification, with the size of the predicted set reflecting the degree of decision ambiguity. We extend the learning-to-defer framework to multiple treatments via a novel \textit{greatest Lower Bound} method, and introduce \textit{conformal policy learning}, which bridges the gap between unobserved ground-truth optimal treatments and estimated optimal treatment rules. Drawing on insights from the noisy-label literature, we develop a randomness-injection approach that guarantees marginal coverage without requiring assumptions on underlying black-box optimal treatment rules. Through experiments on synthetic data and a real-world application to In-Vitro Fertilization (IVF), we demonstrate that our methods produce robust and actionable policies that naturally incorporate clinical considerations while effectively balancing performance and reliability.

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.

We prove a general lower bound for differentially private federated learning protocols with arbitrary public-transcript interactions. The protocol may use any number of adaptive rounds, and each client's local samples may be reused across rounds. For parameter estimation under squared \(\ell_2\) loss, we establish a federated van Trees lower bound for every estimator satisfying a total clientwise sample-level zero-concentrated differential privacy (zCDP) constraint. The main technical ingredient is a privacy-information contraction inequality for complete public transcripts. We illustrate the bound through applications to mean estimation, linear regression, and nonparametric regression.

We introduce FLUXtrapolation, a benchmark for extrapolating ecosystem fluxes under progressively harder distribution shifts. Ecosystem fluxes are central to understanding the carbon, water, and energy cycles, yet they can only be measured directly at sparsely located measurement towers. Producing global flux estimates therefore requires training models on observed sites using globally available covariates and predicting in unobserved regions, that is, upscaling. Flux upscaling is a challenging domain generalization problem that is affected by a shift in covariate distribution across climates, ecosystem types, and environmental conditions, as well as by conditional shift: important drivers remain unobserved at global scale. We provide a quantitative analysis of both these shifts in $P_X$ and $P_{Y\mid X}$. FLUXtrapolation is designed based on domain expertise on flux upscaling: it defines temporal, spatial, and temperature-based extrapolation scenarios and evaluates performance across held-out domains, temporal aggregations, and tail errors. In a pilot study, we find that baselines perform similarly under median hourly RMSE, but separate under the proposed tail-focused and multi-scale evaluation. FLUXtrapolation therefore poses a realistic and thus relevant challenge for machine learning methods under distribution shift; at the same time, progress on this benchmark would directly support the scientific goal of improving flux upscaling.

Mathematical models are invaluable for understanding and predicting how biological systems behave, although their construction requires specifying mechanisms and relationships that are often not perfectly known. In the presence of multiple competing models, model uncertainty should be accounted for when performing inference based on available data. Bayesian model selection is a framework for testing mechanistic hypotheses and generating predictions under model uncertainty, which generally requires computation of the model evidence. In this work, we investigate the reliability of evidence computation methods when parameter non-identifiability -- the inability to distinguish between parameter values given available data -- is present, and find that deterministic evidence approximations can produce misleading model selection results because their underlying assumptions are violated. We propose a novel implementation of adaptive multiple importance sampling for evidence estimation, and demonstrate its robustness against non-identifiability. We use ecological case studies to demonstrate how simple model selection methods fail to produce accurate results, whereas our method yields model selection results that are comparable to those obtained by Markov chain Monte Carlo methods at substantially lower computational cost. Given the pervasiveness of parameter non-identifiability in mathematical biology, this work provides a practical approach to reliable model selection in the presence of poorly identified parameters.

We investigate the global optimization of the objective function arising in continuous sparse regression, specifically the Beurling LASSO (BLASSO), over the space of measures. While Conic Particle Gradient Descent (CPGD) methods are computationally efficient, they may become trapped in local minima due to the non-convexity of the parameterization. To overcome this limitation, we introduce Fast Spawn\&Prune (FS\&P), a stochastic algorithm that extends FastPart introduced in De Castro et al. (2025) and combines CPGD with a birth-death process. The birth mechanism ensures asymptotic global exploration by introducing particles in regions where first-order optimality conditions are violated, while the death process preserves computational efficiency by pruning non-informative particles. We provide the first theoretical guarantee of global convergence for this class of discrete-time stochastic algorithms, without requiring exponentially large initializations. Furthermore, we derive explicit convergence rates for the excess risk, which scale as $\mathcal{O}\big(\left(\log K / K\right)^{\frac{1}{2(2+d)}}\big)$, where $K$ denotes the number of iterations and d the dimension of the domain, thereby quantifying the trade-off between global exploration and local refinement. Moreover, the sample complexity is $\mathcal{O}\big(N^{-\frac{1}{4(2+d)}}\big)$ (up to logarithmic factors). We also propose a horizon-free variant that does not require prior knowledge of the iteration budget.

Classical multivariate statistical methods such as covariance estimation and principal component analysis are well understood mathematically, yet their application at extreme data scales remains challenging. When the number of observations reaches billions, performance is limited by data movement, input-output bottlenecks, and numerical stability rather than arithmetic complexity. This work presents a case study of scaling classical multivariate statistics on a single multi-GPU node. Using C++ and CUDA, a GPU-accelerated workflow was developed to compute sufficient statistics in a single pass over a 10-billion-row dataset. Column sums and cross-product matrices are used to enable downstream computation of means, covariance, correlation, and principal component analysis without revisiting the raw data. The results highlight the importance of data representation, validation using known invariants, and careful numerical treatment when applying established statistical methods at large scale.

Bayesian structural equation modelling (BSEM) offers many advantages such as principled uncertainty quantification, small-sample regularisation, and flexible model specification. However, the Markov chain Monte Carlo (MCMC) methods on which it relies are computationally prohibitive for the iterative cycle of specification, criticism, and refinement that careful psychometric practice demands. We present INLAvaan, an R package for fast, approximate Bayesian SEM built around the Integrated Nested Laplace Approximation (INLA) framework for structural equation models developed by Jamil & Rue (2026, arXiv:2603.25690 [stat.ME]). This paper serves as a companion manuscript that describes the architectural decisions and computational strategies underlying the package. Two substantive applications -- a 256-parameter bifactor circumplex model and a multilevel mediation model with full-information missing-data handling -- demonstrate the approach on specifications where MCMC would require hours of run time and careful convergence work. In constrast, INLAvaan delivers calibrated posterior summaries in seconds.

Markov chain Monte Carlo (MCMC) methods remain the mainstay of Bayesian estimation of structural equation models (SEM), though they often incur a high computational cost. We present a bespoke approximate Bayesian approach to SEM, drawing on ideas from the integrated nested Laplace approximation (INLA, Rue et al., 2009, J. R. Stat. Soc. Series B Stat. Methodol.) framework. We implement a simplified Laplace approximation that efficiently profiles the posterior density in each parameter direction while correcting for asymmetry, allowing for parametric skew-normal estimation of the marginals. Furthermore, we apply a variational Bayes correction to shift the marginal locations, thereby better capturing the posterior mass. Essential quantities, including factor scores and model-fit indices, are obtained via an adjusted Gaussian copula sampling scheme. For normal-theory SEM, this approach offers a highly accurate alternative to sampling-based inference, achieving near-'maximum likelihood' speeds while retaining the precision of full Bayesian inference.

We propose a neural network model for contextual regression in which the regression model depends on contextual features that determine the active submodel and an algorithm to fit the model. The proposed simple contextual neural network (SCtxtNN) separates context identification from context-specific regression, resulting in a structured and interpretable architecture with fewer parameters than a fully connected feed-forward network. We show mathematically that the proposed architecture is sufficient to represent contextual linear regression models using only standard neural network components. Numerical experiments are provided to support the theoretical result, showing that the proposed model achieves lower excess mean squared error and more stable performance than feed-forward neural networks with comparable numbers of parameters, while larger networks improve accuracy only at the cost of increased complexity. The results suggest that incorporating contextual structure can improve model efficiency while preserving interpretability.

For approximating a target distribution given only its unnormalized log-density, stochastic gradient-based variational inference (VI) algorithms are a popular approach. For example, Wasserstein VI (WVI) and black-box VI (BBVI) perform gradient descent in measure space (Bures-Wasserstein space) and parameter space, respectively. Previously, for the Gaussian variational family, convergence guarantees for WVI have shown superiority over existing results for black-box VI with the reparametrization gradient, suggesting the measure space approach might provide some unique benefits. In this work, however, we close this gap by obtaining identical state-of-the-art iteration complexity guarantees for both. In particular, we identify that WVI's superiority stems from the specific gradient estimator it uses, which BBVI can also leverage with minor modifications. The estimator in question is usually associated with Price's theorem and utilizes second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient. On the flip side, WVI can be made more widely applicable by using the reparametrization gradient, which requires only gradients of the log-density. We empirically demonstrate that the use of Price's gradient is the major source of performance improvement.

Arnold Zellner published a seminal paper on Bayes' theorem as an optimal information processing rule, a result that led to the variational formulation of Bayes' theorem, and a central idea in generalized variational inference. Almost 40 years later, we revisit these ideas, but from the perspective of information deletion. We investigate rules that update a posterior distribution into an antedata distribution when a portion of data is removed. In such context, a rule that does not destroy or create nonexistent information is called the optimal information deletion rule and we prove that it coincides with the leave-data-out posterior from Bayes' theorem.

We study in-context learning for nonparametric regression with $α$-Hölder smooth regression functions, for some $α>0$. We prove that, with $n$ in-context examples and $d$-dimensional regression covariates, a pretrained transformer with $Θ(\log n)$ parameters and $Ω\bigl(n^{2α/(2α+d)}\log^3 n\bigr)$ pretraining sequences can achieve the minimax optimal rate of convergence $O\bigl(n^{-2α/(2α+d)}\bigr)$ in mean squared error. Our result requires substantially fewer transformer parameters and pretraining sequences than previous results in the literature. This is achieved by showing that transformers are able to approximate local polynomial estimators efficiently by implementing a kernel-weighted polynomial basis and then running gradient descent.

In this paper, we develop a comprehensive asymptotic and bootstrap theory for checkerboard-based estimation of lower and upper tail copulas under unknown marginal distributions. The estimator is constructed via local bilinear (checkerboard) interpolation of the empirical copula and extended to the tail region to obtain nonparametric estimators of extremal dependence. We first establish almost sure uniform consistency of the checkerboard-smoothed copula estimator by decomposing the error into a stochastic empirical process term and a deterministic approximation bias induced by the checkerboard projection. Under mild growth conditions on the grid size, the estimator is shown to be strongly consistent. Next, we derive weak convergence of the centered and scaled checkerboard copula process in $\ell^\infty([0,1]^2)$, showing that the smoothing does not affect the first-order limit. The resulting Gaussian process coincides with that of the empirical copula, augmented by terms arising from marginal estimation. These results extend to the lower and upper tail copula processes, yielding functional central limit theorems and asymptotic normality of the tail dependence coefficient. Since the limiting covariance depends on unknown tail features and partial derivatives rendering direct inference infeasible, we propose a direct multiplier bootstrap adapted to the checkerboard structure. We prove conditional weak convergence of the bootstrap process to the same limit, ensuring valid inference for smooth functionals. Finally, we illustrate the bootstrap methodology through simulations and statistical applications, including goodness-of-fit testing and inference on tail dependence under a range of dependence structures, demonstrating accurate finite-sample performance.

Although conformal prediction provides robust marginal coverage guarantees, achieving reliable conditional coverage for specific inputs remains challenging. While exact distribution-free conditional coverage is impossible with finite samples, recent work has focused on improving the conditional coverage of standard conformal procedures. Distinct from approaches that target relaxed notions of conditional coverage, we directly target the mean squared error of conditional coverage by refining the quantile regression components that underpin many conformal methods. Leveraging a Taylor expansion, we derive a sharp surrogate objective for quantile regression: a density-weighted pinball loss, where the weights are given by the conditional density of the nonconformity score evaluated at the true quantile. We propose a three-headed quantile network that estimates these weights via finite differences using auxiliary quantile levels at $1-α\pm δ$, subsequently fine-tuning the central quantile by optimizing the weighted loss. We provide a theoretical analysis with exact non-asymptotic guarantees characterizing the resulting excess risk. Extensive experiments on diverse high-dimensional real-world datasets demonstrate remarkable improvements in conditional coverage performance.

Real-world measurements often comprise a dominant signal contaminated by a noisy background. Robustly estimating the dominant signal in practice has been a fundamental statistical problem. Classically, mixture models have been used to cluster the heterogeneous population into homogeneous components. Modeling such data with fully parametric models risks bias under misspecification, while fully nonparametric approaches can dissipate power and computational resources. We propose a middle path: a semiparametric method that models only the dominant component parametrically and leaves the background completely nonparametric, yet remains computationally scalable and statistically robust. So instead of outlier downweighting, traditionally done in robust statistics literature, we maximize the observed likelihood such that the noisy background is absorbed by the nonparametric component. Computationally, we propose a new approximate FFT-accelerated likelihood maximization algorithm. Empirically, this FFT plug-in achieves order-of-magnitude speedups over vanilla weighted EM while preserving statistical accuracy and large sample properties.

We extend a heuristic method for automatic dimensionality selection, which maximizes a profile likelihood to identify "elbows" in scree plots. Our extension enables researchers to make automatic choices of multiple hyper-parameters simultaneously. To facilitate our extension to multi-dimensions, we propose a "softened" profile likelihood. We present two distinct parameterizations of our solution and demonstrate our approach on elastic nets, support vector machines, and neural networks. We also report a small simulation study to investigate violations to an assumption we make, and briefly discuss applications of our method to other data-analytic tasks than hyper-parameter selection.

Copulas are a fundamental tool for modelling multivariate dependencies in data, forming the method of choice in diverse fields and applications. However, the adoption of existing models for multimodal and high-dimensional dependencies is hindered by restrictive assumptions and poor scaling. In this work, we present methods for modelling copulas based on the principles of diffusions and flows. We design two processes that progressively forget inter-variable dependencies while leaving dimension-wise distributions unaffected, provably defining valid copulas at all times. We show how to obtain copula models by learning to remember the forgotten dependencies from each process, theoretically recovering the true copula at optimality. The first instantiation of our framework focuses on direct density estimation, while the second specialises in expedient sampling. Empirically, we demonstrate the superior performance of our proposed methods over state-of-the-art copula approaches in modelling complex and high-dimensional dependencies from scientific datasets and images. Our work enhances the representational power of copula models, empowering applications and paving the way for their adoption on larger scales and more challenging domains.

We study a class of nonconvex-nonconcave minimax problems in which the inner maximization problem satisfies a local Kurdyka-Lojasiewicz (KL) condition that may vary with the outer minimization variable. In contrast to the global KL or Polyak-Lojasiewicz (PL) conditions commonly assumed in the literature -- which are significantly stronger and often too restrictive in practice -- this local KL condition accommodates a broader range of practical scenarios. However, it also introduces new analytical challenges. In particular, as an optimization algorithm progresses toward a stationary point of the problem, the region over which the KL condition holds may shrink, resulting in a more intricate and potentially ill-conditioned landscape. To address this challenge, we show that the associated maximal function is locally generalized Hölder smooth. Leveraging this key property, we develop an inexact proximal gradient method for solving the minimax problem, where the inexact gradient of the maximal function is computed by applying a proximal gradient method to a KL-structured subproblem. Under mild assumptions, we establish complexity guarantees for computing an approximate stationary point of the minimax problem.

The rank correlation ξ(X,Y), recently established by Sourav Chatterjee and already popular in the statistics literature, takes values in [0,1], where 0 characterizes independence of X and Y, and 1 characterizes perfect dependence of Y on X. Unlike concordance measures such as Spearman's ρ, which capture the degree of positive or negative dependence, ξquantifies the strength of functional dependence. In this paper, we study the attainable set of pairs (ξ(X,Y),ρ(X,Y)). The resulting ξ-\r{ho}-region is a convex set whose boundary is characterized by a novel family of absolutely continuous, asymmetric copulas having a diagonal band structure. Moreover, we prove that ξ(X,Y)\leq|ρ}(X,Y)| whenever Y is stochastically increasing or decreasing in X, and we identify the maximal difference ρ(X,Y)-ξ(X,Y) as exactly 0.4. Our proofs rely on a convex optimization problem under various equality and inequality constraints, as well as on ordering properties for ξand ρ. Our results contribute to a better understanding of Chatterjee's rank correlation, which typically yields substantially smaller values than Spearman's ρwhen quantifying positive dependencies. In particular, when interpreting the values of Chatterjee's rank correlation on the scale of ρ, the quantity \sqrtξ appears to be more appropriate.

This work studies and develop projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings-that include the design of the variant (i). Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank-Wolfe algorithm within our model class can have a regret guarantee better than O(T^3/4) (T is the time horizon) without additional assumptions, that the current algorithms do not have optimal constants, that the algorithm benefits from similar anytime properties O(t^3/4) not requiring to know T in advance, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.

We provide a simple distribution regression estimator for treatment effects in the difference-in-differences (DiD) design. Our procedure is particularly useful when the treatment effect differs across the distribution of the outcome variable. Our proposed estimator easily incorporates covariates and, importantly, can be extended to settings where the treatment potentially affects the joint distribution of multiple outcomes. Our key identifying restriction is that the untreated outcome distribution does not exhibit an interaction effect of group and time. This assumption results in a parallel trend assumption on a transformation of the distribution. We highlight the relationship between our procedure and assumptions with the changes-in-changes approach of Athey and Imbens (2006). We also reexamine the Card and Krueger (1994) study of the impact of minimum wages on employment to illustrate the utility of our approach.

High-dimensional linear regression has been thoroughly studied in the context of independent and identically distributed data. We propose to investigate high-dimensional regression models for independent but non-identically distributed data. To this end, we suppose that the set of observed predictors (or features) is a random matrix with a variance profile and with dimensions growing at a proportional rate. Assuming a random effect model, we study the predictive risk of the ridge estimator for linear regression with such a variance profile. In this setting, we provide deterministic equivalents of this risk and of the degree of freedom of the ridge estimator. For certain class of variance profile, our work highlights the emergence of the well-known double descent phenomenon in high-dimensional regression for the minimum norm least-squares estimator when the ridge regularization parameter goes to zero. We also exhibit variance profiles for which the shape of this predictive risk differs from double descent. The proofs of our results are based on tools from random matrix theory in the presence of a variance profile that have not been considered so far to study regression models. Numerical experiments are provided to show the accuracy of the aforementioned deterministic equivalents on the computation of the predictive risk of ridge regression. We also investigate the similarities and differences that exist with the standard setting of independent and identically distributed data.

The ability to maintain and manipulate information over time is a fundamental aspect of living beings and Artificial Intelligence. While modern models have achieved remarkable success in tasks like natural language processing, evaluating the capacity of novel architectures to process sequential information remains computationally expensive and time-consuming. Testing a new architecture often requires scaling up to massive datasets and models, leading to vast computational costs and slow iteration cycles. In this paper, we propose CogScale, a benchmark of 14 scalable synthetic tasks designed to isolate and evaluate specific cognitive and memory abilities at different parametrizable scales. By providing a standardized, lightweight framework, CogScale allows researchers to rapidly validate architectural innovations before committing to large-scale training. To establish a solid baseline, we evaluate seven distinct architectures: Gated Recurrent Unit (GRU), Long Short-Term Memory (LSTM), xLSTM, Echo State Network (ESN), Mamba, Transformer Decoder, and Transformer Encoder-Decoder. These evaluations are conducted under strict parameter budgets (1k, 10k, and 100k) and across different difficulty levels and scales. Our results show that while classical RNNs and Echo State Networks excel at basic retention within strict parameter budgets, only attention mechanisms and modern state-space models consistently maintain high performance as reasoning complexity and task difficulty scale.

Through case studies, we demonstrate how multiverse analysis can strengthen the robustness and transparency of computational social science findings against alternative methodological decisions. We conduct multiverse analyses of three published social science studies that use the following computational methods: Bayesian analysis, network generative modeling, and machine learning with or without large language models. These methods are applied frequently in computational social science studies, yet entail a greater degree of arbitrariness in terms of methodological choices, or "researcher degrees of freedom." Our multiverse analyses reveal how the empirical findings in these studies vary as a function of various plausible decision combinations. Our three case studies also expose an often-ignored motivation for conducting multiverse analysis: Showing which methodological combinations lead to computational failure. These failed cases are usually not communicated in the published reports, even though these sophisticated computational methods have a much higher likelihood of failure. We end our paper with suggestions on how to find defensible decision combinations for multiverse analysis of computational social science studies and how to communicate multiverse analysis findings fairly.

Inference about treatment effects for time-to-event outcomes is often obscured by the presence of competing events. A particularly complex situation arises when the treatment influences the occurrence of the competing event. A comprehensive assessment should then account for different mechanisms by which the treatment and the competing event together produce the apparent treatment effect. Here, we propose a decomposition of the treatment's effect on the event of interest (target), characterising how it arises due to four distinct mechanisms involving both the target and competing events. Based on a causal model, the decomposition relies on cross-world estimands reflecting counterfactual scenarios in which the treatment affects the two events as if set to conflicting levels. We specify exchangeability and consistency assumptions under which the decomposition can be estimated from observed data. We discuss how the new decomposition reveals the role of the competing event and serves as a basis for defining causally interpretable estimands in the presence of competing events. Finally, we demonstrate the use of the four-way decomposition with datasets from two randomised trials.

Accurately assessing financial risk requires capturing both individual asset volatility and the complex, asymmetric dependence structures that emerge during extreme market events. While modern diffusion-based models have advanced multivariate forecasting, they often suffer from a "normality bias" when trained end-to-end, sacrificing marginal calibration for joint coherence and consistently underestimating tail risk. To address this, we propose a Diffusion-Copula framework that explicitly decouples the learning of marginal distributions from their dependence structure. We employ deep Mixture Density Networks to capture heavy-tailed asset dynamics, followed by a Classification-Diffusion Copula to model the joint dependence. Applied to cryptocurrency markets, our approach demonstrates superior performance over state-of-the-art baselines in forecasting systemic extremes of both marginal and joint events. Crucially, we demonstrate that while baseline models classify simultaneous market crashes as statistically impossible "Black Swans" (high surprise), our framework identifies them as "Expected Crashes" (low surprise), successfully preserving the correlation structure necessary for robust risk management during contagion events.

We study the problem of estimating a monotone function $f:\{0,1\}^d\to[0,1]$ from noisy observations at uniformly random vertices of the Boolean hypercube. As a measure of complexity for the target~$f$, we use the total $L^1$-influence $I(f)=\sum\_{i=1}^d(\E[f(X)\mid X\_i=1]-\E[f(X)\mid X\_i=0])$, a classical quantity in Boolean analysis that is nonnegative for monotone functions and controls the effective dimensionality of the estimation problem: through a spectral concentration result in the spirit of Friedgut's junta theorem, the Fourier spectrum of any $f$ with $I(f)\leqslant K$ concentrates on low-degree subsets of the influential coordinates. We establish minimax bounds over the class $\cF\_K=\{f:\{0,1\}^d\to[0,1],\;f\text{ monotone},\; I(f)\leqslant K\}$: \[ c\,\frac{K^2}{(\log n)^{3/2}} \;\leqslant\; \inf\_{\hat f}\;\sup\_{f\in\cF\_K}\; \E\bigl[\|\hat f - f\|\_2^2\bigr] \;\leqslant\; C\,\frac{K}{\sqrt{\log n}}, \] where $n$ is the sample size. The upper bound holds for all $K\geqslant 1$ and is uniform in the ambient dimension~$d$ (under the mild condition $\log d\leqslant n^{1-\varepsilon}$). It is achieved by a Fourier thresholding estimator that adapts to the unknown~$K$. The lower bound relies on a Varshamov--Gilbert packing on the middle layer of the hypercube combined with Fano's inequality.

Stochastic gradient methods are central to modern large-scale learning, but their use with incomplete covariates remains delicate since imputation schemes generally introduce systematic gradient biases, as shown for linear models. In this work, we prove that all parametric models exhibit similar gradient bias for various imputation procedures and characterize exactly the dependence on the missingness ratio vector $p$, with $O(\|p\|)$ as the leading term. We exploit this analysis to propose a simple debiasing procedure for stochastic gradient descent (SGD) with missing values based on Richardson extrapolation, which leverages the exact expression of the gradient bias. The key idea is to \emph{deliberately add missingness}: from an already incomplete observation, we generate a further-thinned version at a higher, controlled missingness level, and combine the two resulting stochastic gradients to cancel the leading bias term. We prove that one Richardson step reduces the gradient bias from $O(\|p\|)$ to $O(\|p\|^2)$ under several missingness scenarios. Our proposed method is computationally efficient, model-agnostic and applies to any parametric loss whose stochastic gradient can be computed after imputation. Furthermore, when missing indicators are independent, the population gradient bias is a multilinear polynomial in $p$ and depends only on population gradient errors induced by declaring a single coordinate missing. In this case, our method generalizes to a multi-step Richardson procedure which recursively cancels higher-order terms. Empirically, Richardson debiasing improves optimization and estimation across several generalized linear models and combines positively with widely used imputation procedures such as MICE. These results suggest that, somewhat counter-intuitively, adding controlled missingness on top of existing missing data can make stochastic learning from incomplete data more accurate.

In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.

Matrix-structured parameters frequently appear in many artificial intelligence models such as large language models. More recently, an efficient Muon optimizer is designed for matrix parameters of large-scale models, and shows markedly faster convergence than the vector-wise algorithms. Although some works have begun to study convergence properties (i.e., optimization error) of the Muon optimizer, its generalization properties (i.e., generalization error) is still not established. Thus, in this paper, we study generalization error of the Muon optimizer based on algorithmic stability and mathematical induction, and prove that the Muon has a generalization error of $O\big(\frac{1}{Nκ^{T}}\big)$, where $N$ is training sample size, and $T$ denotes iteration number, and $κ>0$ denotes minimum difference between singular values of gradient estimate. To enhance generalization of the Muon, we propose an effective mixed Muon (MiMuon) optimizer by cautiously using orthogonalization of gradient, which is a hybrid of Muon and momentum-based SGD optimizers. Then we prove that our MiMuon optimizer has a lower generalization error of $O\big(\frac{1}{N}\big)$ than $O\big(\frac{1}{Nκ^{T}}\big)$ of Muon optimizer, since $κ$ generally is very small. Meanwhile, we also studied the convergence properties of our MiMuon algorithm, and prove that our MiMuon algorithm has the same convergence rate of $O(\frac{1}{T^{1/4}})$ as the Muon algorithm. Some numerical experimental results on training large models including Qwen3-0.6B and YOLO26m demonstrate efficiency of the MiMuon optimizer.

Validating interpretable surrogate models for ensemble learners requires measuring agreement between the ensemble's internal representation and its surrogate approximation, rather than mere association. Correlation-based approaches are scale-invariant and fail to detect systematic discrepancies in co-occurrence structure. We propose a statistical framework grounded in the agreement-association distinction, centered on the normalized Loss of Interpretability (nLoI). Rooted in the Cressie-Read power divergence family with lambda equal to 2, the nLoI admits a closed-form decomposition into within-node and between-node components, providing a unique diagnostic capability to identify precisely where and why reconstruction fails. The framework incorporates four complementary measures capturing distinct structural facets of approximation quality. A unified permutation testing procedure delivers valid inference for all measures within a single resampling pass. Theoretical properties, including boundedness and symmetry, are established for each metric. Monte Carlo simulations and empirical evaluations confirm exact Type I error control and demonstrate that these measures detect reconstruction fidelity gradients invisible to correlation-based alternatives. The framework is developed and illustrated in the context of Explainable Ensemble Trees (E2Tree), and empirical evaluation on three benchmark datasets illustrates the practical utility of the framework.

We investigate the asymptotic properties of the Lévy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the Lévy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.

Roll-call data analysis aims to estimate legislators' ideal points and quantify the associated uncertainty. Existing approaches either rely on Bayesian methods implemented via Markov chain Monte Carlo sampling or focus primarily on point estimation, with uncertainty typically assessed through resampling procedures such as the bootstrap. Consequently, the computational burden of these approaches can become substantial when applied to large roll-call datasets. To address this challenge, we propose a computationally efficient likelihood method for estimating ideal points and their standard errors. Leveraging the Pólya--Gamma identity, we develop a variational expectation--maximization algorithm for estimating ideal points and introduce a variational Louis' method to approximate the observed Fisher information for standard error estimation. Numerical studies and applications to U.S. congressional roll-call data demonstrate that the proposed method produces accurate ideal point estimates and reliable standard errors while being substantially more computationally efficient than existing approaches.

We study an online market-making problem in which a learner sequentially posts bid and ask prices for a single asset while interacting with traders holding private valuations. Unlike existing online learning formulations that assume fully censored feedback, we introduce an action-dependent feedback model inspired by real limit order books: when a trade occurs, the trader's valuation remains hidden, whereas when no trade occurs, informative feedback about supply and demand is revealed. We show that this additional information fundamentally changes the learnability of the problem. In the stochastic setting with i.i.d. market prices, we propose an elimination-based algorithm that achieves $O(\sqrt T)$ regret with high probability, without requiring any smoothness assumptions on the distribution of trader valuations. We then extend this result to a broad class of mean-reverting price processes by considering both local, autoregressive dynamics and a weaker global drift condition based on cumulative deviations from the mean. Under either assumption, we establish high-probability $O(\sqrt T)$ regret bounds, relying on a new concentration inequality of independent interest. Finally, in the adversarial setting with oblivious prices, we design an explore-then-perturb algorithm that guarantees $O(T^{2/3})$ regret in expectation. Our results quantify the value of observing the order book in online market making and demonstrate that even limited, action-dependent feedback can substantially improve regret guarantees compared to standard bandit feedback models.

We study post-hoc Learning to Defer (L2D) through the lens of ideal distributions: divergence-regularized reweightings of the data distribution under which a model attains low loss. We define deferral via the density-ratio between a model's and an expert's ideals. Using the reduction from density-ratio estimation to class-probability estimation, we derive the DR CPE losses for post-hoc L2D scorers. Deferral decisions are then made by thresholding the scorer, allowing deferral rates to be adjusted without retraining. For KL-based ideal distributions, our deferral rules recovers Chow's rule under the original distribution and a connection to an expert-tilted Bayes posterior -- which incorporates the expert's performance -- depending on if the ideal distributions are joint or marginal distributions. Experimentally, our approach is competitive compared to common baselines and more robust across dataset settings. More broadly, our results cast post-hoc L2D as density-ratio learning between ideal distributions, bridging Chow-style rules, expert comparison, and elucidating connections to related learning settings including anomaly detection.

Linear regression is widely used to model relationships between responses and predictors. In modern applications, one encounters data where the responses are non-Euclidean random objects situated in a metric space, paired with Euclidean predictors. Global Fréchet regression generalizes linear regression to such general settings, however statistical inference has remained largely unexplored. We develop a significance test for the null hypothesis that the Fréchet regression function does not depend on the predictors, addressing the challenge of an absence of linear operations in metric spaces. We also develop a test for the partial effect of a subset of the predictors in analogy to, but quite different from, the partial F-tests commonly used in classical linear regression under Gaussian assumptions. Key ideas are to employ random multipliers to obtain non-degenerate null distributions for the proposed test statistics and the Cauchy combination method. We obtain consistency and convergence results under the null hypothesis and contiguous alternatives and demonstrate the finite sample performance of the proposed tests through simulations on network data represented by graph Laplacians and spherical data with geodesic distances. We further illustrate our method using transport networks arising from New York City taxi trip data and U.S. energy source compositional data.

This study investigates the macroeconomic determinants and dynamic behaviour of personal remittances as a share of Gross Domestic Product (GDP) in Nepal, emphasizing external demand in major destination countries and domestic monetary policy. Using annual data (1993-2024), we construct composite indices via Principal Component Analysis (PCA) for multi-country external demand and a domestic Monetary Conditions Index (MCI). Our small-sample econometric pipeline includes Autoregressive Distributed Lag (ARDL) bounds testing, Engle-Granger cointegration, Dynamic OLS (DOLS), and a two-step Error Correction Model (ECM). We also employ Granger causality tests and multi-model forecasting using machine learning and ECM scenarios. The analysis reveals a strong positive long-run effect of external demand on remittances and a significant negative impact of tighter domestic monetary conditions. The ECM confirms a stable cointegrating relationship, correcting approximately 26% of disequilibria annually. Medium-term projections indicate remittances will remain structurally important, reaching around 28.3% of GDP by 2030 under baseline conditions, while exhibiting high sensitivity to external demand shocks. This study advances the literature by integrating PCA-derived external demand and monetary conditions indices within a unified ARDL-ECM framework for small samples. Focusing on one of the world's most remittance-dependent economies, it offers actionable insights for monetary policy calibration, migration diversification, and the productive utilization of remittance inflows.

Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise, transforming it into a simple prior, and then use denoising score matching, a consequence of Tweedie's formula, to learn the score function and generate clean samples from noise. However, non-Gaussian diffusion models with state-dependent diffusion coefficient have been largely underexplored, as have the corresponding Tweedie's formulae. In this work, we extend Tweedie's formula to important non-Gaussian processes, including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes, thereby yielding the corresponding denoising score-matching objectives. We then apply the derived formulae to image and financial time series generation using GBM- and CIR-based diffusion models, and to empirical Bayes estimation under the BESQ setting. The reported experimental results demonstrate the potential of non-Gaussian models.

Testing for Hardy-Weinberg equilibrium (HWE) is a fundamental component of genetic data analysis, widely used for quality control and model validation. Although HWE testing is well established for autosomal loci, inference on the X chromosome is more complex due to sex-specific genotype structures and potential sex differences in minor allele frequency (sdMAF). Existing tests differ in their assumptions about sdMAF and male sample inclusion, often leading to distinct but poorly characterized null hypotheses. We develop a general statistical framework for HWE inference using the robust allele-based regression model. By formulating HWE testing as an assessment of allele-level dependence, the framework directly parameterizes Hardy-Weinberg disequilibrium, unifies existing Pearson chi-square-based tests under explicit modeling assumptions, and clarifies their null hypotheses, degrees of freedom, and sensitivity to sdMAF. The framework also accommodates covariate and population-structure adjustment within a unified regression-based formulation. The proposed framework provides robust, interpretable, and flexible inference, establishing a unified statistical foundation for HWE testing across autosomal and X-chromosomal regions. Simulation studies and analysis of high-coverage 1000 Genomes Project data demonstrate that commonly used X-chromosome tests can exhibit inflated type I error or misleading inference when sdMAF is present.

We study high-dimensional inference in correlated two-view models, focusing on spectral methods for strong detection and weak recovery. We introduce a general framework, motivated by a TAP type heuristic from statistical physics, that provides a unified treatment of three canonical models: high-dimensional canonical correlation analysis, and the correlated spiked Wigner and Wishart models. Our main contribution is to construct explicit spectral algorithms in all three settings, that achieve strong detection and weak recovery down to the corresponding thresholds, where we prove matching information-theoretic lower bounds. Furthermore, our spectral procedures operate without knowledge of the model parameters, relying solely on the observed data. This demonstrates the optimality of spectral methods in these models and the broad statistical applicability of the framework.

Hallucination remains a central failure mode of large language models, but existing benchmarks operationalize it inconsistently across summarization, question answering, retrieval-augmented generation, and agentic interaction. This fragmentation makes it unclear whether a mitigation that works in one setting reduces hallucinations across contexts. Current benchmarks either require human annotation and fixed references that may be memorized, or rely on observations in settings that are difficult to reproduce. To study root causes, we introduce HalluWorld, an extensible benchmark grounded in an explicit reference-world formulation: a model hallucinates when it produces an observable claim that is false with respect to this world. Building on this view, we construct synthetic and semi-synthetic environments in which the reference world is fully specified, the model's view is controlled, and hallucination labels are generated automatically. HalluWorld spans gridworlds, chess, and realistic terminal tasks, enabling controlled variation of world complexity, observability, temporal change, and source-conflict policy, and disentangling hallucinations into fine-grained error categories. We evaluate frontier and open-weight language models across these settings and find consistent patterns: perceptual hallucination on directly observed information is near-solved for frontier models, while multi-step state tracking and causal forward simulation remain difficult and are not generally solved by extended thinking. In the terminal setting, models also struggle with when to abstain. The uneven profile of failures across probe types and domains suggests that hallucinations arise from distinct failure modes rather than a single capability. Our results suggest that controlled reference worlds offer a scalable and reproducible path toward measuring and reducing hallucinations in modern language models.

In complex multivariate systems, interactions among variables are defined by dependency structures, often encoded as directed acyclic graphs ($\text{DAGs}$). However, dependency structures can vary across subjects, and ignoring this structural heterogeneity introduces bias and obscures subpopulation-specific dependencies. To address this, we propose Directed Acyclic Graph-based Dependency Clustering via Alternating Direction Method of Multipliers (DAG-DC-ADMM), a unified framework built upon Structural Equation Modeling (SEM) that jointly learns cluster assignments and cluster-specific dependency structures. We encode acyclicity via a smooth constraint and integrate a groupwise truncated Lasso fusion penalty (gTLP) to cluster subjects based on their structural similarity. This yields a nonconvex optimization problem that incorporates sparsity, acyclicity, and structural consensus constraints. We address the nonconvexity by using the augmented Lagrangian method and solve it with an adapted version of the Alternating Direction Method of Multipliers (ADMM) for difference-of-convex programs. For certain graph structures, such as upper triangular adjacency matrices, our algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point. Experiments demonstrate that our method recovers cluster-specific causal dependency structures with a high true positive rate and a low false discovery rate. This capability enables the robust discovery of heterogeneous dependencies across subjects where the subpopulation label is unknown.

Stochastic gradient descent (SGD) is a fundamental optimization algorithm widely used in modern machine learning. In this paper, we propose Factor-Augmented SGD (FSGD), a new optimization method that leverages latent factor representations in high-dimensional learning tasks. Unlike standard two-stage dimension reduction approaches that rely on offline representation learning and full data storage, a key novelty of FSGD is that it operates purely on streaming data, making it scalable to large-scale and high-dimensional problems. Furthermore, we establish the first theoretical framework that explicitly incorporates latent factor estimation error into the analysis of SGD, and provide moment convergence in $\ell^s$ norm under decaying step sizes and mini-batch updates. Our results provide a new foundation for employing SGD reliably and scalably in high-dimensional machine learning systems.

Sequential prediction is challenging in regimes of delayed disambiguation, where early observations are ambiguous and multiple latent explanations remain plausible until sufficient evidence accumulates. Standard approaches based on marginal inference struggle in this setting, either collapsing uncertainty prematurely or failing to recover once informative evidence arrives. We introduce EviTrack, a test-time inference framework that operates over latent trajectories rather than marginal states. EviTrack maintains a set of competing trajectory hypotheses and applies evidence- and likelihood-ratio-based selection to delay commitment until supported by data, drawing inspiration from hypothesis management in multiple hypothesis tracking and track-before-detect. To evaluate this setting, we construct a controlled synthetic benchmark with known latent ground truth that explicitly exhibits delayed disambiguation. At matched inference budget, EviTrack substantially outperforms sampling-based baselines, achieving faster post-disambiguation recovery. These results show that, in delayed disambiguation regimes, moderate trajectory-level selection is more effective than increasing sampling coverage, highlighting selection over sampling as a key principle for reliable sequential inference.

Can researchers use local open-weight models instead of commercial APIs for LLM text classification? Local models avoid marginal API charges, keep data on the researcher's machine, and make exact model versions easier to preserve. I benchmark five local models against four commercial API models on 34 political science classification tasks. Local models are often competitive, especially on simpler tasks. In a task-specific oracle comparison, local models match or exceed API performance on 9 tasks; on average, the best API model exceeds the best local model by 0.015 F1. The four strongest observed model means fall within 0.021 F1. API models have their clearest edge on complex tasks with many labels or multiple outputs per item. Batching several items in one prompt usually reduces local runtime per item, but specific model-task pairs can return invalid response formats or labels. Taken together, the results make local open-weight models a practical candidate alternative for many political science classification tasks, provided researchers validate candidate models on task-specific labels and check batching reliability before scaling up.

We introduce DeRegiME -- Deep Regime Mixture of Experts -- a direct multi-horizon probabilistic forecaster that separates latent uncertainty regimes from the underlying signal and softly assigns each forecast location to learned recurring regimes using a sparse variational Gaussian process (GP) whose nonstationary regime-mixing kernel and Student-t likelihood combine per-regime sub-kernels and noise processes via a shared gate. This yields a single sparse-GP posterior, not a mixture of GP experts. DeRegiME addresses a key limitation of neural forecasters: point forecasts discard residual uncertainty, and probabilistic heads -- whether single marginals, uninterpreted mixtures, quantile sets, or diffusion samples -- rarely expose the regime structure of the residual. Yet distribution shift in noisy heteroskedastic time series may be abrupt, gradual, or horizon-dependent and often appears in residual uncertainty rather than the conditional mean. DeRegiME yields an interpretable mean-residual-noise decomposition with a direct-sum feature-space representation that anchors regimes as clusters of residual similarity whose transitions surface as implicit changepoints. The effective number of regimes is pruned by the stick-breaking gate. We prove kernel validity and predictive-density propriety, and across ten benchmarks and three encoder grids DeRegiME improves negative log predictive density (NLPD) by 20.3% over the strongest encoder-matched baseline, a DeepAR/GluonTS-style dynamic Student-t head, with parallel gains on CRPS (3.0%) and MSE (4.7%). Improvements are consistent across all datasets, which span abrupt, gradual, and seasonal shifts.

Physical activity (PA) plays an important role in maintaining and improving health. Daily steps have been a key PA measure that is easily accessible with common wearable devices. However, methods are lacking to recommend a personalized optimal distribution of daily steps over a period of time for the best of certain health biomarkers. In this paper, we fill this void based on the data from the All of Us Research Program which includes months of step counts as well as repeated measurements of key health biomarkers. We develop a new offline reinforcement learning (RL) algorithm to learn personalized and optimal PA distributions associated with cardiometabolic risk, where the action is a function representing the daily step distribution over a period of time. Simulation studies demonstrate the advantage of the proposed approach over existing continuous-action RL methods. The learned optimal policy from the All of Us data generally suggests people take more daily steps and also follow a more consistent pattern of PA over time while offering tailored recommendations for subgroups in blood glucose level, body mass index, blood pressure, age, and sex.

The construction of models from data is a significant contributor to the energetic costs of computation. Because of this, understanding how foundational thermodynamic bounds apply to modeling algorithms will be increasingly important. Here, we study the thermodynamic costs of a basic and fundamental modeling algorithm: simple linear regression. Following Landauer, we approximate the thermodynamic lower bound on irreversibly performing both exact linear regression and linear regression via stochastic gradient descent as implemented on floating-point numbers. From this, we derive energycost aware scaling laws for the optimal dataset size for training a linear regression model given a generalization error dependent demand for inference. Additionally, we discuss a method to lower bound the entropy production from the mismatch cost for algorithms with continuous input variables.

This paper generalises inference functions (Godambe, 1960) to distributional statistical models, in which each probability measure is represented by a distribution--kernel pair $(T_θ, φ) \in \mathcal S'(\mathbb R) \times \mathcal S(\mathbb R)$. The generalisation is strategically motivated: the key properties of maximum likelihood estimation-consistency and asymptotic normality -derive not from maximising the likelihood but from the MLE being the root of a regular inference function. Extending inference functions to the distributional setting provides an optimality theory for models lacking classical densities or finite moments. The extension requires enlarging the notion of observation. We introduce observation operators $\mathcal O : \mathcal S'(\mathbb R) \to \mathcal Y$ mapping distributional models to an observation space, and define inference functionals as estimating equations composed with these operators. The framework encompasses classical point observations, interval-censored data, convolutional measurements, and transform-based statistics. We establish asymptotic theory (consistency, asymptotic normality, Godambe optimality) under mild conditions and derive a hierarchy of information bounds -- classical Fisher information dominates the information available through the observation operator, which in turn dominates the information captured by any inference functional -- via the Hájek--Le~Cam convolution theorem. The two gaps quantify distinct sources of information loss: the observation mechanism and the choice of inference functional. Examples include sinusoidal inference functions for heavy-tailed distributions, interval-censored location inference, elliptically contoured models, and nuisance parameters via the Bhapkar--Godambe projection.

We derive the asymptotic distribution of the spatial Cram'{e}r--von Mises statistic for testing bivariate independence in stationary random fields on $\mathbb{R}^2$ under polynomial $β$-mixing dependence, and document the Python implementation that reproduces all simulation results. The classical test assumes i.i.d. observations; we extend it to spatially dependent data by combining three ingredients: (i) a Davydov-type covariance bound yielding integrability of the spatial covariance kernel under $θ> 2(2+δ)/δ$; (ii) a reformulation of the inner-form test statistic as a degenerate U-statistic of order~2 with product kernel $Q = G_1 \otimes G_2$, following De Wet (1980); and (iii) an extension of Gregory's (1977) U-statistic limit theorem to $β$-mixing sequences via Yoshihara (1976). The limit distribution is a weighted sum of correlated $χ^2_1$ variables whose eigenvalues factor as products of marginal eigenvalues; in the small-bandwidth limit the correlation vanishes and the limit reduces to the classical i.i.d. form. Explicit eigenvalue formulas are given for three weight functions (uniform, optimal normal, Anderson--Darling), producing computable critical values. The software generates Mat'{e}rn random fields by circulant embedding, computes the test statistic via the inner-form kernel decomposition, evaluates asymptotic critical values by Monte Carlo, and runs permutation-based alternatives. Simulation experiments show that the Anderson--Darling weight achieves the best power, while the Mantel and cross-$K$ tests have no power against cross-dependence in spatially correlated fields.

Tensor-valued data arise naturally in neuroimaging, genomics, climate science, and spatiotemporal networks, where multilinear dependencies across modes carry information that is destroyed under vectorization. Existing approaches either impose a single low-rank structure, which can miss localized signal, or treat the tensor as a long vector, which discards its multiway geometry. We propose a *Dual-Channel Tensor Neural Network* (DC-TNN) that decomposes each tensor input into a low-rank core and a sparse refinement, and processes the two components through coupled neural channels. The framework is structure-agnostic and accommodates CP, Tucker, and tensor-train cores within a single architecture. For estimation, we establish non-asymptotic risk bounds for the DC-TNN estimator that decompose into network approximation, core estimation, and refinement-selection terms, and show that the effective dimension is determined jointly by the core rank and refinement sparsity rather than by the ambient tensor size. For inference, we develop a *structure-aware conformal ROC* procedure that calibrates within the core-refinement latent space and produces ROC and AUC confidence bands with finite-sample, distribution-free coverage. Building on this, we propose a *conformal structure selector* that, to our knowledge, is the *first distribution-free procedure* for choosing among candidate tensor decompositions with finite-sample validity. Simulations and an analysis of a protein dataset demonstrate competitive predictive accuracy, reliable uncertainty quantification, and consistent recovery of the tensor structure.

Many clinical risk scores are deployed as additive rules with nonnegative integer points assigned to relevant binary predictive features. These integer weights not only make the score easier to use in practice but also promote sparsity in the resulting prediction model. Such risk scores are often derived by first fitting a regression model and then rounding the estimated coefficients to the nearest integer after appropriate scaling. This approach is computationally fast but does not guarantee optimality of the resulting score. Alternatively, one may search over all possible integer weights to directly optimize a value function by posing the problem as an integer programming task. However, the associated computational burden can be substantial, especially when the value function is nonconcave or even discontinuous. In this paper, we develop new machine learning algorithms that employ a flexible greedy optimization strategy to learn such additive scoring directly under explicit and sensible optimality objectives. We apply the proposed method to a large electronic health record (EHR) cohort in Epic Cosmos to construct an integer-weighted comorbidity score for measuring the risk of post-discharge mortality. We also conduct a simulation study to examine the finite-sample operating characteristics.

In this article, we present $\textbf{ldmppr}$, an R package for estimating, evaluating, simulating from, and visualizing location-dependent marked spatial point processes. To date, it has commonly been assumed that the marks associated with a point process are independent of the locations. However, when dealing with many point processes, such as those arising in forestry applications, the independence assumption proves unreasonable. We introduce a practical framework for generating marked point processes with dependence between the marks and locations. We provide a brief discussion of the theory underpinning our modeling approach and outline the use of the package in a typical scenario involving real data. We highlight the functionality of the package for both generating from and assessing the goodness-of-fit of a given model, enabling users to generate realistic point patterns given a reference pattern or parameter values of interest.

Two widely used randomized algorithms are the sketch-and-solve method for least-squares regression and the randomized SVD for low-rank approximation. These algorithms apply a random embedding to compress a target matrix, and they perform computations on the compressed matrix to save computational cost. This paper asks, what is the optimal random embedding in these algorithms? Also, what is the sharpest possible error bound for the optimal embedding? The paper proves that a random orthonormal matrix is minimax optimal for the sketch-and-solve algorithm while any rotation-invariant embedding is minimax optimal for the randomized SVD. Following these results, the paper obtains the best possible error bounds for sketched least-squares and the randomized SVD. Last, empirical experiments provide evidence of universality phenomena, in which several random embeddings lead to similar accuracy to the optimal embeddings in practice.

Schedule-Free Learning has shown promise as a practical anytime training method for machine learning, showing success across dozens of standard benchmark problems. However, strong performance for LLM training has only been demonstrated at small scales. We identify a number of fixes necessary to scale up Schedule-Free Learning to larger batch sizes and model sizes, and present a learning-rate-free and schedule-free method (ScheduleFree+) for training large language models which greatly outperforms Warmup-Stable-Decay (WSD) schedules. We also demonstrate that Schedule-Free Learning is most effective for long duration training, and at 1000 tokens per parameter, it outperforms SOTA schedules by 31%. Schedule-Free Learning provides a theoretical foundation for the use of model averaging and checkpoint merging during pretraining.

Latent Class Analysis (LCA) is widely used to identify unobserved subgroups in social and behavioural sciences. A long-standing challenge for LCA is the interpretability of the latent classes, due to the high complexity of the estimated item response probability matrix. To address this, we propose a computationally efficient post-estimation refinement procedure that enhances model interpretability by a sparse model estimate. The method begins by estimating a classical, unrestricted, latent class model and determining the number of classes using the Bayesian information criterion (BIC). It is followed by a refinement step that further performs model selection on the item-specific response probabilities based on the initial estimate. This refinement penalises the number of distinct response probability levels per item, collapsing redundant levels to yield a sparse matrix that is significantly easier to interpret than those produced by classical LCA. We provide asymptotic theory showing that the proposed procedure consistently recovers the sparse pattern of the item response probabilities for each item, and further validate its performance through extensive simulations. The practical power of the proposed method is further illustrated via an application to survey data on social role performance, where it provides a parsimonious and clear characterisation of the resulting latent classes. The code for implementing the proposed method is publicly available at https://github.com/florence07/Sparse-LCA-Refinement.

Split conformal prediction provides finite-sample marginal coverage under exchangeability, but this guarantee averages over the random calibration sample. We study instead the law of the calibration-conditional coverage induced by a realized conformal threshold. In the continuous i.i.d. setting this law is exactly $Beta(k,n+1-k)$, so the usual marginal guarantee corresponds to its mean. We take this beta law as a finite-sample reference object and quantify departures from it using Wasserstein distances on $[0,1]$. The framework yields direct bounds on marginal coverage gaps and on bad-calibration probabilities, and separates different sources of non-i.i.d. behavior according to how they deform the beta reference: test-side shift acts through a transport map on the coverage scale, while calibration dependence changes the order-statistic law itself. We instantiate the framework in scale-shift, clustered, and stationary mixing settings, where the induced deformations can be characterized explicitly or through Berry-Esseen approximations. Simulations on dependent processes confirm that the first-order approximation tracks the empirical Wasserstein distance even at moderate sample sizes.

The U.S. social safety net delivers essential services at mass scale, but access burdens persist, as congested contact or call centers serve as a primary mode of application completion and assistance. In Holmes v. Knodell, Missouri's SNAP call centers were so congested that nearly half of all application denials were procedural, caused by applicants' inability to complete required interviews, rather than underlying ineligibility. The judge ruled these system failures led to a violation of procedural due process. We propose a performance evaluation framework based on queueing models from operations research and management to assess and improve access in such systems. Operational access failures of call centers are distinct from prior automation failures in benefits provision. Emergent arbitrariness arises from interactions between system dynamics and access demand, rather than from an explicit algorithmic rule, making diagnosis and repair inherently system-level. We develop a queueing model that incorporates phenomena that distinguish social services from standard service domains, redials and abandonment, through which backlogs generate endogenous congestion. Standard queueing guidance from Erlang-A that does not address endogenous congestion fundamentally understaffs, which could lead to persistent shortfalls in practice. Using a fluid approximation, we derive steady-state performance metrics to analytically characterize the impacts of bundled staffing and service delivery changes. We fit model parameters to call-center data disclosed in court documents. Our queueing model can support ex-ante evaluation and design of access systems, inform policy levers for improving access, and provide evidence about whether applicants are afforded a meaningful opportunity to be served at scale.

Given an observation $\mathbf Y \in \mathbb{R}^{d_1\times d_2}$ from the model $\mathbf Y = \mathbf X + \mathbf E$ where $\mathbf X$ is constant and $\mathbf E$ has i.i.d. $N(0,1)$ entries, we consider the problem of detecting a planted submatrix in the mean matrix $\mathbf X$. Specifically, we aim to distinguish the null hypothesis $\mathbf X = 0$ from the alternative hypothesis in which $\mathbf X$ is non-zero only on a submatrix of size $s_1 \times s_2$ with elevated entries bounded below by $μ>0$. We establish a minimax lower bound characterizing how large $μ$ must be to ensure that the two hypotheses are distinguishable with high probability. Furthermore, we derive novel minimax-optimal tests achieving the lower bound, and describe extensions of these tests that are adaptive to unknown sparsity levels $s_1$ and $s_2$. In contrast with previous work, which required restrictive assumptions on $s_1,s_2, d_1$ and $d_2$, our non-asymptotic upper and lower bounds match for any configuration of these parameters.

The main XAI attribution methods for deep neural networks -- GradCAM, SHAP, LIME, Integrated Gradients -- operate on separate theoretical foundations and are not formally comparable. We present GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), a mathematical framework establishing a representation theory for attributions: every additive, linear, and continuous attribution functional on L^2(Q,mu) admits a unique canonical representation (Q, w, Delta), proved necessary by the Riesz Representation Theorem. This class encompasses SHAP, IG, LIME and linearized GradCAM, but excludes nonlinear functionals such as standard GradCAM or attention maps. Seven formal theorems provide simultaneous guarantees absent in any individual method: (T1) necessary canonical form; (T2) exact completeness; (T3) Monte Carlo convergence O(1/sqrt(m))+O(1/k); (T4) exact Shapley Interaction Values; (T5) Hoeffding ANOVA decomposition; (T6) Sobol sensitivity generalization; (T7) multi-scale extension (MS-GRALIS) with minimum-variance weights. An algebraic appendix justifies the GRALIS-SIV correspondence via the Mobius transform without circularity. GRALIS satisfies 13.5/14 axiomatic properties vs. 2.5-6/14 for individual methods, including completeness, sensitivity, locality, order-k interactions and optimal multi-scale aggregation simultaneously. Preliminary validation on BreaKHis (1,187 histology images, DenseNet-121) reports deletion faithfulness AUC +0.015 (malignant), 96% class-conditional consistency, SAL = 0.762+/-0.109 and sparsity index 0.39. Extended comparison with baseline XAI methods is planned for a companion paper.

We introduce entropic strict minimum message length (SMML), a risk-sensitive generalization of strict minimum message length coding. The proposed criterion replaces expected two-part codelength under the prior predictive distribution with an exponential certainty equivalent, thereby defining a one-parameter family of coding rules that interpolates between Bayesian average-case coding and worst-case minimax coding. We show that ordinary SMML is recovered in the risk-neutral limit, while the extreme risk-sensitive limit yields a minimax codelength criterion; when centered by the oracle maximum likelihood codelength, this criterion coincides with the normalized maximum likelihood (NML) minimax-regret principle. We further prove that entropic SMML admits a variational characterization as a Kullback--Leibler-regularized worst-case expected codelength, giving it a PAC--Bayes-type interpretation. We establish a joint asymptotic theory linking the sample size $n$ and the risk parameter $τ$, showing that in regular parametric models the transition between Bayesian, robust, and minimax coding regimes occurs on a logarithmic scale. For regular exponential families, the fixed-codebook partition remains affine in sufficient-statistic space, while the codepoints satisfy a tilted moment-matching condition and admit an interpretation as tilted Bregman centroids. These results position entropic SMML as an information-theoretic bridge between MML, PAC--Bayes, and MDL.

ANOVA Simultaneous Component Analysis (ASCA) is the current state-of-theart chemometric tool for analyzing and interpreting high-dimensional experimental data from a Design of Experiment (DoE). Being a multivariate extension of the ANOVA, ASCA makes a perfect tandem with DoE. This tutorial review recommends best practices for using ASCA, building upon the long-established combination of ANOVA and DoE theory developed over the last century. These recommendations are grounded in a comprehensive literature review and illustrated through a guiding example.

Masked diffusion large language models (dLLMs) are a promising alternative to autoregressive generation. While reinforcement learning (RL) methods have recently been adapted to dLLM fine-tuning, their objectives typically depend on sequence-level marginal likelihoods, which are intractable for masked diffusion models. To address this, we derive Discrete Tilt Matching (DTM), a likelihood-free method that recasts dLLM fine-tuning as state-level matching of local unmasking posteriors under reward tilting. DTM takes the form of a weighted cross-entropy objective with explicit minimizer, and admits control variates that improve training stability. On a synthetic maze-planning task, we analyze how DTM's annealing schedule and control variates affect training stability and prevent mode collapse. At scale, fine-tuning LLaDA-8B-Instruct with DTM yields strong gains on Sudoku and Countdown while remaining competitive on MATH500 and GSM8K.

Model-based approaches for (bio)process systems often suffer from incomplete knowledge of the underlying physical, chemical, or biological laws. Universal differential equations, which embed neural networks within differential equations, have emerged as powerful tools to learn this missing physics from experimental data. However, neural networks are inherently opaque, motivating their post-processing via symbolic regression to obtain interpretable mathematical expressions. Genetic algorithm-based symbolic regression is a popular approach for this post-processing step, but provides only point estimates and cannot quantify the confidence we should place in a discovered equation. We address this limitation by applying Bayesian symbolic regression, which uses Reversible Jump Markov Chain Monte Carlo to sample from the posterior distribution over symbolic expression trees. This approach naturally quantifies uncertainty in the recovered model structure. We demonstrate the methodology on a Lotka-Volterra predator-prey system and then show how a well-designed experiment leads to lower uncertainty in a fed-batch bioreactor case study.

Treatment effects estimated from a randomized controlled trial are local not only to the study population but also to the time at which the trial was conducted. The literature on generalizing experimental findings to new populations is extensive, yet transporting effects across time has received far less attention, and even defining the target estimand is nonobvious. We formalize the transported average treatment effect under a separable temporal effects assumption, derive two identification strategies: replicated trials and common arm, and develop doubly robust, semiparametrically efficient estimators for each. Applied to a large archive of headline A/B tests, the common arm strategy is substantially more precise but exhibits systematic bias when the temporal factor depends on the gap between intervention and measurement rather than on measurement time alone, while the replicated trials strategy, which allows this dependence, tracks the ground truth more faithfully. Simulation studies investigate when each strategy is reliable and when it silently fails.

The asymptotic properties of multivariate Szász-Mirakyan estimators for cumulative distribution functions (cdf) supported on the nonnegative orthant are investigated. Explicit bias and variance expansions are derived on compact subsets of the interior, yielding sharp mean squared error characterizations and optimal smoothing rates. The analysis shows that the proposed Poisson smoothing yields a non-negligible variance reduction relative to the empirical cdf, leading to asymptotic efficiency gains that can be quantified through local and global deficiency measures. The behavior of the estimator near the boundary of its support is examined separately. Under a boundary-layer scaling that preserves nondegenerate Poisson smoothing as the evaluation point approaches the boundary of $[0,\infty)^d$, bias and variance expansions are obtained that differ fundamentally from those in the interior region. In particular, the variance reduction mechanism disappears at leading order, implying that no asymptotically optimal smoothing parameter exists in the boundary regime. Central limit theorems and almost sure uniform consistency are also established. Together, these results provide a unified asymptotic theory for multivariate Szász-Mirakyan cdf estimation and clarify the distinct roles of smoothing in the interior and boundary regions.

We propose Q-learning with Adjoint Matching (QAM), a novel TD-based reinforcement learning (RL) algorithm that tackles a long-standing challenge in continuous-action RL: efficient optimization of an expressive diffusion or flow-matching policy with respect to a parameterized Q-function. Effective optimization requires exploiting the first-order information of the critic, but it is challenging to do so for flow or diffusion policies because direct gradient-based optimization via backpropagation through their multi-step denoising process is numerically unstable. Existing methods work around this either by only using the value and discarding the gradient information, or by relying on approximations that sacrifice policy expressivity or bias the learned policy. QAM sidesteps both of these challenges by leveraging adjoint matching, a recently proposed technique in generative modeling, which transforms the critic's action gradient to form a step-wise objective function that is free from unstable backpropagation, while providing an unbiased, expressive policy at the optimum. Combined with temporal-difference backup for critic learning, QAM consistently outperforms prior approaches on hard, sparse reward tasks in both offline and offline-to-online RL.

Estimating the unknown reward functions driving agents' behaviors is of central interest in inverse reinforcement learning and game theory. To tackle this problem, we develop a unified framework for reward function recovery in two-player zero-sum matrix games and Markov games with entropy regularization, where we aim to reconstruct the underlying reward functions given observed players' strategies and actions. This task is challenging due to the inherent ambiguity of inverse problems, the non-uniqueness of feasible rewards, and limited observational data coverage. To address these challenges, we establish the reward function's identifiability using the quantal response equilibrium (QRE) under linear assumptions. Building upon this theoretical foundation, we propose a novel algorithm to learn reward functions from observed actions. Our algorithm works in both static and dynamic settings and is adaptable to incorporate different methods, such as Maximum Likelihood Estimation (MLE). We provide strong theoretical guarantees for the reliability and sample efficiency of our algorithm. Further, we conduct extensive numerical studies to demonstrate the practical effectiveness of the proposed framework, offering new insights into decision-making in competitive environments.

In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $β$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.

Estimating latent epidemic states and model parameters from partially observed, noisy data remains a major challenge in infectious disease modeling. State-space formulations provide a coherent probabilistic framework for such inference, yet fully Bayesian estimation is often computationally prohibitive because evaluating the observed-data likelihood requires integration over a latent trajectory. The Sequential Monte Carlo squared (SMC$^2$) algorithm offers a principled approach for joint state and parameter inference, combining an outer SMC sampler over parameters with an inner particle filter that estimates the likelihood up to the current time point. Despite its theoretical appeal, this nested particle filter imposes substantial computational cost, limiting routine use in near-real-time outbreak response. We propose Ensemble SMC$^2$ (eSMC$^2$), a computationally efficient variant that replaces the inner particle filter with an Ensemble Kalman Filter (EnKF) to approximate the incremental likelihood at each observation time. While this substitution introduces bias via a Gaussian approximation, we mitigate finite-sample effects using an unbiased Gaussian density estimator and adapt the EnKF for epidemic data through state-dependent observation variance. This makes our approach particularly suitable for overdispersed incidence data commonly encountered in infectious disease surveillance. Simulation experiments with known ground truth and an application to 2022 United States (U.S.) monkeypox incidence data demonstrate that eSMC$^2$ achieves substantial computational gains while producing posterior estimates comparable to SMC$^2$. The method accurately reconstructs epidemic trajectories and estimates key epidemiological parameters, providing an efficient framework for sequential Bayesian inference from imperfect surveillance data.

Online bilevel optimization (OBO) is a powerful framework for machine learning problems where both outer and inner objectives evolve over time, requiring dynamic updates. Current OBO approaches rely on deterministic \textit{window-smoothed} regret minimization, which may not accurately reflect system performance when functions change rapidly. In this work, we introduce a novel search direction and show that both first- and zeroth-order (ZO) stochastic OBO algorithms leveraging this direction achieve sublinear {stochastic bilevel regret without window smoothing}. Beyond these guarantees, our framework enhances efficiency by: (i) reducing oracle dependence in hypergradient estimation, (ii) updating inner and outer variables alongside the linear system solution, and (iii) employing ZO-based estimation of Hessians, Jacobians, and gradients. Experiments on online parametric loss tuning and black-box adversarial attacks validate our approach.

Vine copulas offer flexible multivariate dependence modeling and have become widely used in machine learning. Yet, structure learning remains a key challenge. Early heuristics, such as Dissmann's greedy algorithm, are still considered the gold standard but are often suboptimal. We propose random search algorithms and a statistical framework based on model confidence sets, to improve structure selection, provide theoretical guarantees on selection probabilities and excess risk, as well as serve as a foundation for ensembling. Empirical results on real-world data sets show that our methods consistently outperform state-of-the-art approaches.

Understanding the dynamics of feature learning in neural networks (NNs) remains a significant challenge. The work of (Mousavi-Hosseini et al., 2023) analyzes a multiple index teacher-student setting and shows that a two-layer student attains a low-rank structure in its first-layer weights when trained with stochastic gradient descent (SGD) and a strong regularizer. This structural property is known to reduce sample complexity of generalization. Indeed, in a second step, the same authors establish algorithm-specific learning guarantees under additional assumptions. In this paper, we focus exclusively on the structure discovery aspect and study it under weaker assumptions, more specifically: we allow (a) NNs of arbitrary size and depth, (b) with all parameters trainable, (c) under any smooth loss function, (d) tiny regularization, and (e) trained by any method that attains a second-order stationary point (SOSP), e.g.\ perturbed gradient descent (PGD). At the core of our approach is a key $\textit{derandomization}$ lemma, which states that optimizing the function $\mathbb{E}_{\mathbf{x}} \left[g_θ(\mathbf{W}\mathbf{x} + \mathbf{b})\right]$ converges to a point where $\mathbf{W} = \mathbf{0}$, under mild conditions. The fundamental nature of this lemma directly explains structure discovery and has immediate applications in other domains including an end-to-end approximation for MAXCUT, and computing Johnson-Lindenstrauss embeddings.

We consider a process $X^\ve$ solution of a stochastic Volterra equation with an unknown parameter $θ^\star$ in the drift function. The Volterra kernel is singular near zero, exhibiting a behavior comparable to $K\_0(u)=cu^{α-1} \id{u>0}$ with $α\in (1/2,1)$.It is assumed that the diffusion coefficient is proportional to $\ve \to 0$. Based on discrete observations, with a mesh size $h\to0$, of the Volterra process, we construct a Quasi Maximum Likelihood Estimator. The main step is to assess the error arising in the reconstruction of the path of a semimartingale from the inversion of the Volterra kernel. We show that this error decreases as $h^{1/2}$ regardless of the value of $α$. Then, we can introduce an explicit contrast function, which yields an efficient estimator when $\ve \to 0$.

The task of learning a diffusion-based neural sampler for drawing samples from an unnormalized target distribution can be viewed as a stochastic optimal control problem on path measures. However, the training of neural samplers can be challenging when the target distribution is multimodal with significant barriers separating the modes, potentially leading to mode collapse. We propose a framework named Proximal Diffusion Neural Sampler (PDNS) that addresses these challenges by tackling the stochastic optimal control problem via proximal point method on the space of path measures. PDNS decomposes the learning process into a series of simpler subproblems that create a path gradually approaching the desired distribution. This staged procedure traces a progressively refined path to the desired distribution and promotes thorough exploration across modes. For a practical and efficient realization, we instantiate each proximal step with a proximal weighted denoising cross-entropy (WDCE) objective. We demonstrate the effectiveness and robustness of PDNS through extensive experiments on both continuous and discrete sampling tasks, including challenging scenarios in molecular dynamics and statistical physics. Our code is available at https://github.com/AlexandreGUO2001/PDNS.

This paper proposes two algorithms for estimating square Wasserstein distance matrices from a small number of entries. These matrices are used to compute manifold learning embeddings like multidimensional scaling (MDS) or Isomap, but contrary to Euclidean distance matrices, are extremely costly to compute. We analyze matrix completion from upper triangular samples and Nyström completion in which $\mathcal{O}(d\log(d))$ columns of the distance matrices are computed where $d$ is the desired embedding dimension, prove stability of MDS under Nyström completion, and show that it can outperform matrix completion for a fixed budget of sample distances. Finally, we show that classification of the OrganCMNIST dataset from the MedMNIST benchmark is stable on data embedded from the Nyström estimation of the distance matrix even when only 10\% of the columns are computed.

Nonstationary spatial processes can often be represented as stationary processes on a warped spatial domain. Selecting an appropriate spatial warping function for a given application is often difficult and, as a result of this, warping methods have largely been limited to two-dimensional spatial domains. In this paper, we introduce a novel approach to modeling nonstationary, anisotropic spatial processes using neural autoregressive flows (NAFs), a class of invertible mappings capable of generating complex, high-dimensional warpings. Through simulation studies we demonstrate that a NAF-based model has greater representational capacity than other commonly used spatial process models. We apply our proposed modeling framework to a subset of the 3D Argo Floats dataset, highlighting the utility of our framework in real-world applications.

We study the problem of model aggregation within the Wasserstein space for probability measures on the real line. Given a fixed finite collection of candidate probability models, we consider the associated class of Wasserstein barycenters and develop a data-driven calibration framework in which the aggregation weights are statistically learned from empirical information associated with a target distribution. From a variational perspective based on $Γ$-convergence, we establish consistency of the resulting aggregation scheme, showing that empirical minimizers converge to the minimizers of the actual problem, along with the associated barycentric estimators, under mild conditions. The performance of the proposed method is evaluated through synthetic experiments and illustrated on a real dataset from a temperature monitoring network of sensors.

We mathematically analyze and numerically study an actor-critic machine learning algorithm for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) partial differential equations from stochastic control theory. The architecture of the critic (the estimator for the value function) is structured so that the boundary condition is always perfectly satisfied (rather than being included in the training loss) and utilizes a biased gradient which reduces computational cost. The actor (the estimator for the optimal control) is trained by minimizing the integral of the Hamiltonian over the domain, where the Hamiltonian is estimated using the critic. We show that the training dynamics of the actor and critic neural networks converge in a Sobolev-type space to a certain infinite-dimensional ordinary differential equation (ODE) as the number of hidden units in the actor and critic $\rightarrow \infty$. Further, under a convexity-like assumption on the Hamiltonian, we prove that any fixed point of this limit ODE is a solution of the original stochastic control problem. This provides an important guarantee for the algorithm's performance in light of the fact that finite-width neural networks may only converge to a local minimizers (and not optimal solutions) due to the non-convexity of their loss functions. In our numerical studies, we demonstrate that the algorithm can solve stochastic control problems accurately in up to 200 dimensions. In particular, we construct a series of increasingly complex stochastic control problems with known analytic solutions and study the algorithm's numerical performance on them. These problems range from a linear-quadratic regulator equation to highly challenging equations with non-convex Hamiltonians, allowing us to identify and analyze the strengths and limitations of this neural actor-critic method for solving HJB equations.

Observational cohort data is an important source of information for understanding the causal effects of treatments on survival and the degree to which these effects are mediated through changes in disease-related risk factors. However, these analyses are often complicated by irregular data collection intervals and the presence of longitudinal confounders and mediators. We propose a causal mediation framework that jointly models longitudinal exposures, confounders, mediators, and time-to-event outcomes as continuous functions of age. This framework for longitudinal covariate trajectories enables statistical inference even at ages where the subject's covariate measurements are unavailable. The observed data distribution in our framework is modeled using an enriched Dirichlet process mixture (EDPM) model. Using data from the Atherosclerosis Risk in Communities cohort study, we apply our methods to assess how medication -- prescribed to target cardiovascular disease (CVD) risk factors -- affects the time-to-CVD death.

Modern industrial systems are often subject to multiple failure modes, and their conditions are monitored by multiple sensors, generating multiple time-series signals. Additionally, time-to-failure data are commonly available. Accurately predicting a system's remaining useful life (RUL) requires effectively leveraging multi-sensor time-series data alongside multi-mode failure event data. In most existing models, failure modes and RUL prediction are performed independently, ignoring the inherent relationship between these two tasks. Some models integrate multiple failure modes and event prediction using black-box machine learning approaches, which lack statistical rigor and cannot characterize the inherent uncertainty in the model and data. This paper introduces a unified approach to jointly model the multi-sensor time-series data and failure time concerning multiple failure modes. This proposed model integrate a Cox proportional hazards model, a Convolved Multi-output Gaussian Process, and multinomial failure mode distributions in a hierarchical Bayesian framework with corresponding priors, enabling accurate prediction with robust uncertainty quantification. Posterior distributions are effectively obtained by Variational Bayes, and prediction is performed with Monte Carlo sampling. The advantages of the proposed model is validated through extensive numerical and case studies with jet-engine dataset.

We study risk-sensitive reinforcement learning in finite discounted MDPs with recursive entropic risk measures (ERM), where the risk parameter $β\neq 0$ controls the agent's risk attitude: $β>0$ for risk-averse and $β<0$ for risk-seeking behavior. A generative model of the MDP is assumed to be available. Our focus is on the sample complexities of learning the optimal state-action value function (value learning) and an optimal policy (policy learning) under recursive ERM. We introduce a model-based algorithm, called Model-Based ERM $Q$-Value Iteration (MB-RS-QVI), and derive PAC-type bounds on its sample complexity for both value and policy learning. Both PAC bounds scale exponentially with $|β|/(1-γ)$, where $γ$ is the discount factor. We also establish corresponding lower bounds for both value and policy learning, showing that exponential dependence on $|β|/(1-γ)$ is unavoidable in the worst case. The bounds are tight in the number of states and actions ($S$ and $A$), providing the first rigorous sample complexity guarantees for recursive ERM across both risk-averse and risk-seeking regimes.

The motivation of this article is to improve inferences on the covariation in environmental exposures, motivated by data from a study of Toddlers Exposure to SVOCs in Indoor Environments (TESIE). The challenge is that the sample size is limited, so empirical covariance provides a poor estimate. In related applications, Bayesian factor models have been popular; these approaches express the covariance as low rank plus diagonal and can infer the number of factors adaptively. However, they have the disadvantage of shrinking towards a diagonal covariance, often under estimating important covariation patterns in the data. Alternatively, the dimensionality problem is addressed by collapsing the detailed exposure data within chemical classes, potentially obscuring important information. We apply a feature aware covariance regression extension of Bayesian factor analysis, which improves performance by including information from features summarizing properties of the different exposures. This approach enables shrinkage to more flexible covariance structures, reducing the over-shrinkage problem, as we illustrate in the TESIE data using various chemical features.

Given an unnormalized probability density $π\propto\mathrm{e}^{-V}$, estimating its normalizing constant $Z=\int_{\mathbb{R}^d}\mathrm{e}^{-V(x)}\mathrm{d}x$ or free energy $F=-\log Z$ is a crucial problem in Bayesian statistics, statistical mechanics, and machine learning. It is challenging especially in high dimensions or when $π$ is multimodal. To mitigate the high variance of conventional importance sampling estimators, annealing-based methods such as Jarzynski equality and annealed importance sampling are commonly adopted, yet their quantitative complexity guarantees remain largely unexplored. We take a first step toward a non-asymptotic analysis of annealed importance sampling. In particular, we derive an oracle complexity of $\widetilde{O}\left(\frac{dβ^2{\mathcal{A}}^2}{\varepsilon^4}\right)$ for estimating $Z$ within $\varepsilon$ relative error with high probability, where $β$ is the smoothness of $V$ and $\mathcal{A}$ denotes the action of a curve of probability measures interpolating $π$ and a tractable reference distribution. Our analysis, leveraging Girsanov's theorem and optimal transport, does not explicitly require isoperimetric assumptions on the target distribution. Finally, to tackle the large action of the widely used geometric interpolation, we propose a new algorithm based on reverse diffusion samplers, establish a framework for analyzing its complexity, and empirically demonstrate its efficiency in tackling multimodality.

Gaussian processes (GPs) are flexible, probabilistic, nonparametric models widely used in fields such as spatial statistics and machine learning. A drawback of Gaussian processes is their computational cost, with $O(N^3)$ time and $O(N^2)$ memory complexity, which makes them prohibitive for large data sets. Numerous approximation techniques have been proposed to address this limitation. In this work, we systematically compare the accuracy of different Gaussian process approximations with respect to likelihood evaluation, parameter estimation, and prediction, explicitly accounting for the computational time required. We analyze the trade-off between accuracy and runtime on multiple simulated and large-scale real-world data sets and find that Vecchia approximations consistently provide the best accuracy-runtime trade-off across most settings considered.

Climate change is a critical issue that will be in the political agenda for the next decades. While it is important for this topic to be discussed at higher levels, it is also of paramount importance that the populations became aware of the problem. As different countries may face more or less severe repercussions, it is also useful to understand the degree of awareness of specific populations. In this paper, we present a geographically-informed hierarchical clustering analysis aimed at identify groups of countries with a similar level of climate change awareness. We employ a Ward-like clustering algorithm that combines information pertaining climate change awareness, socio-economic factors, climate-related characteristics of different countries, and the physical distances between countries. To choose suitable values for the clustering hyperparameters, we propose a customized algorithm that takes into account the within-clusters homogeneity, the between-clusters separation and that explicitly compares the geographically-informed and non-geographical partitioning. The results show that the geographically-informed clustering provides more stability of the partitions and leads to interpretable and geographically-compact aggregations compared to a clustering in which the geographical component is absent. In particular, we identify a clear contrast among Western countries, characterized by high and compact awareness, and Asian, African, and Middle Eastern countries having greater variability but still lower awareness.

Many real-world decision problems require solving, again and again, combinatorial optimization instances drawn from a common distribution. A recent line of structured learning methods exploits this regularity by learning policies that pair a statistical model with a tractable combinatorial oracle, instead of solving each instance independently. Training such policies is notoriously difficult, however: the resulting empirical risk is piecewise constant in the model parameters, which hinders gradient-based optimization, and only a few theoretical guarantees have been provided so far. We address this issue by analyzing smoothed (perturbed) policies: adding controlled random perturbations to the direction used by the linear oracle yields a differentiable surrogate risk and improves generalization. Our main contribution is a generalization bound that decomposes the excess risk into $(\mathit{i})$ perturbation bias, $(\mathit{ii})$ statistical estimation error, and $(\mathit{iii})$ optimization error. The perturbation bias is controlled by the \emph{fan-crossing probability}, a new geometric quantity measuring the likelihood that a perturbation changes the oracle solution. We introduce two complementary conditions to bound it--the \emph{Uniformly Bounded Density} (UBD) property, yielding a sharp ${O}(λ)$ bias, and the weaker \emph{Uniform Weak moment} (UW) property, yielding a sub-linear bound--both capturing the geometric interaction between the statistical model and the normal fan of the feasible polytope. The statistical estimation error is controlled via a uniform deviation bound over the policy class, with rate ${O}(1/(λ\sqrt{n}))$ that scales inversely in the smoothing parameter. Concerning the optimization error, we exploit kernel Sum-of-Squares methods to mitigate the curse of dimensionality of global optimization.

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables, where the objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a ``prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of ``prophet inequalities." In this paper we develop a game-theoretic characterization that supports a principled approach for deriving sharp non-asymptotic prophet inequalities for single threshold stopping rules. We demonstrate that sharp constants in the ratio- and difference-type prophet inequalities are determined by the optimal values of infinite two-person zero-sum game on the unit square with particular payoff kernels, while the the solutions to the game provide optimal stopping rules and least favorable distributions. Among other things, this formulation also allows a systematic way to tackle restricted classes of distributions. The proposed framework leads to a numerically efficient algorithmic paradigm that allows computing sharp constants in prophet inequalities with any prescribed level of accuracy.

We give an example of a class of distributions that is learnable up to constant error in total variation distance with a finite number of samples, but not learnable under $(\varepsilon, δ)$-differential privacy with the same target error. This weakly refutes a conjecture of Ashtiani.

Randomized experiments are the gold standard for estimating treatment effects, and randomization serves as a reasoned basis for inference. In widely used stratified randomized experiments, randomization-based finite-population asymptotic theory enables valid inference for the average treatment effect, relying on normal approximation and a Neyman-type conservative variance estimator. However, when the sample size is small or the outcomes are skewed, the Neyman-type variance estimator may become overly conservative, and the normal approximation can fail. To address these issues, we propose a sharp variance estimator and two causal bootstrap methods to more accurately approximate the sampling distribution of the weighted difference-in-means estimator in stratified randomized experiments. The first causal bootstrap procedure is based on rank-preserving imputation and we prove its second-order refinement over normal approximation. The second causal bootstrap procedure is based on constant-treatment-effect imputation and is further applicable in paired experiments. In contrast to traditional bootstrap methods, where randomness originates from hypothetical super-population sampling, our analysis for the proposed causal bootstrap is randomization-based, relying solely on the randomness of treatment assignment in randomized experiments. Numerical studies and two real data applications demonstrate advantages of our proposed methods in finite samples. The \texttt{R} package \texttt{CausalBootstrap} implementing our method is publicly available.

We introduce a novel strategy to train randomised predictors in federated learning, where each node of the network aims at preserving its privacy by releasing a local predictor but keeping secret its training dataset with respect to the other nodes. We then build a global randomised predictor which inherits the properties of the local private predictors in the sense of a PAC-Bayesian generalisation bound. We consider the synchronous case where all nodes share the same training objective (derived from a generalisation bound), and the heterogenous and homogenous cases where each node may have its own personalised training objective. We show through a series of numerical experiments that our approach achieves a comparable predictive performance to that of the batch approach where all datasets are shared across nodes. Moreover the predictors are supported by numerically nonvacuous generalisation bounds while preserving privacy for each node. We explicitly compute the increment on predictive performance and generalisation bounds for our two federated settings, highlighting the price to pay to preserve privacy.

We consider an experimental design setting in which units are assigned to treatment after being sampled sequentially from an infinite population. We derive asymptotic efficiency bounds that apply to data from any experiment that assigns treatment as a (possibly randomized) function of covariates and past outcome data, including stratification on covariates and adaptive designs. For estimating the average treatment effect of a binary treatment, our results show that no further first order asymptotic efficiency improvement is possible relative to an estimator that achieves the Hahn (1998) bound in an experimental design where the propensity score is chosen to minimize this bound. Our results also apply to settings with multiple treatments with possible constraints on treatment, as well as covariate based sampling of a single outcome.

This study demonstrates the existence of a testable condition for the identification of the causal effect of a treatment on an outcome in observational data, which relies on two sets of variables: observed covariates to be controlled for and a suspected instrument. Under a causal structure commonly found in empirical applications, the testable conditional independence of the suspected instrument and the outcome given the treatment and the covariates has two implications. First, the instrument is valid, i.e. it does not directly affect the outcome (other than through the treatment) and is unconfounded conditional on the covariates. Second, the treatment is unconfounded conditional on the covariates such that the treatment effect is identified. We suggest tests of this conditional independence based on machine learning methods that account for covariates in a data-driven way and investigate their asymptotic behavior and finite sample performance in a simulation study. We also apply our testing approach to evaluating the impact of fertility on female labor supply when using the sibling sex ratio of the first two children as supposed instrument, which by and large points to a violation of our testable implication for the moderate set of socio-economic covariates considered.

Traditional randomized A/B experiments assign arms with uniform random (UR) probability, such as 50/50 assignment to two versions of a website to discover whether one version engages users more. To more quickly and automatically use data to benefit users, multi-armed bandit algorithms such as Thompson Sampling (TS) have been advocated. While TS is interpretable and incorporates the randomization key to statistical inference, it can cause biased estimates and increase false positives and false negatives in detecting differences in arm means. We introduce a more Statistically Sensitive algorithm, TS-PostDiff (Posterior Probability of Small Difference), that mixes TS with traditional UR by using an additional adaptive step, where the probability of using UR (vs TS) is proportional to the posterior probability that the difference in arms is small. This allows an experimenter to define what counts as a small difference, below which a traditional UR experiment can obtain informative data for statistical inference at low cost, and above which using more TS to maximize user benefits is key. We evaluate TS-PostDiff against UR, TS, and two other TS variants designed to improve statistical inference. We consider results for the common two-armed experiment across a range of settings inspired by real-world applications. Our results provide insight into when and why TS-PostDiff or alternative approaches provide better tradeoffs between benefiting users (reward) and statistical inference (false positive rate and power). TS-PostDiff's adaptivity helps efficiently reduce false positives and increase statistical power when differences are small, while increasing reward more when differences are large. The work highlights important considerations for future Statistically Sensitive algorithm development that balances reward and statistical analysis in adaptive experimentation.

Microsimulation models used by ministries of finance and central banks rely on parametric processes for lifetime earnings that capture only first and second moments of the conditional distribution and miss long-range nonlinear structure. We propose SAGA, a decoder-only transformer for irregular tabular panel sequences, paired with a split conformal calibration wrapper that delivers individual-level prediction intervals with finite-sample marginal coverage guarantees. Trained on the longitudinal Swedish LISA register over 1990 to 2022, comprising 2,143,817 individuals and 61,284,903 person-years, the model forecasts annual labor earnings at horizons of one to thirty years and aggregates them by Monte Carlo into present-discounted lifetime earnings distributions. Against the canonical Guvenen, Karahan, Ozkan, and Song parametric process and tabular and recurrent baselines, SAGA reduces continuous ranked probability score by 31.9 percent at the ten-year horizon and mean absolute error by 37.7 percent at the twenty-year horizon. Conformal intervals achieve nominal coverage to within 0.4 percentage points marginally and within 2.4 percentage points on the worst-case demographic subgroup. The reconstructed lifetime earnings Gini coefficient is 0.327 against the partially observed truth of 0.341 and the GKOS estimate of 0.378. Model weights, calibration tables, and a synthetic equivalent dataset are released for replication outside the protected SCB MONA environment.

Unobserved confounding is a fundamental challenge for estimating causal effects. To address unobserved confounding, recent literature has turned to two different approaches -- proxy variables and the use of multiple treatments. The first approach, commonly referred to as proximal causal inference, requires proxies to be assigned to specific asymmetric roles: treatment-inducing proxies (negative control exposures), variables that act as common causes of the treatment and outcome, and outcome-inducing proxies (negative control outcomes). In practice, however, identifying variables that satisfy these asymmetric roles can be difficult depending on the application domain. The second approach, commonly referred to as the ``Deconfounder," deals with multiple conditionally independent treatments. There has been limited progress towards developing a consistent estimation method for this setting. As the primary contribution of this work, we establish that causal effects are identifiable in both settings when the unobserved confounder is categorical under suitable conditions. Our approach builds on a mixture learning perspective: we show that the underlying confounding structure can be recovered by identifying the corresponding mixture distribution. We propose an estimation procedure based on tensor decomposition, which allows consistent recovery of the latent structure and comes with non-asymptotic guarantees. Simulation studies and real data experiments demonstrate that the proposed method performs well even with limited data.

Bayesian latent space models offer a principled approach to network representation, but rely on correct specification of both geometry and link function. Real-world networks often violate these assumptions, exhibiting geometric mismatch and structural anomalies that break standard metric properties. We show that such misspecification pushes the data-generating distribution outside the model class, causing Bayesian inference to become overconfident and poorly calibrated. To address this, we propose a generalized posterior framework for random geometric graphs. We introduce Link-Sequential R-SafeBayes, a method that exploits dyadic conditional independence to estimate prequential risk and adaptively tune posterior regularization. Experiments on synthetic and real-world networks demonstrate improved calibration, better link prediction performance, and a reliable criterion for selecting latent geometries across Euclidean, spherical, and hyperbolic spaces.

Structural identifiability analysis determines whether the parameters of a mechanistic ordinary differential equation (ODE) model can be uniquely recovered from ideal observations and is therefore a fundamental prerequisite for reliable parameter estimation. This tutorial presents a modern, reproducible computational framework for symbolic structural identifiability analysis using the Julia package StructuralIdentifiability.jl. We provide a rigorous yet accessible introduction to local and global identifiability, observability, parameter-to-output mappings, and identifiable parameter combinations, together with a unified workflow based on the core functions @ODEmodel, assess_local_identifiability, assess_identifiability, and find_identifiable_functions. The framework is demonstrated through seven case studies from epidemiology, pharmacokinetics, and systems biology, illustrating globally identifiable systems, local-only identifiability, structural non-identifiability, and recovery of identifiability through additional measurements and reparameterization. Beyond the theoretical foundations, the tutorial emphasizes practical model reformulation, experimental design, and reproducible scientific workflows within the Julia SciML ecosystem, providing a comprehensive reference for researchers and graduate students working with mechanistic ODE models.

Decision-makers often deploy the best-performing treatment from a randomized experiment, creating a winner's curse: selection favors treatments whose observed outcomes are high partly because of statistical noise, so the naïve estimate of the winner is upward biased. We distinguish two forms of winner's curse, bias relative to the true best treatment (global) and bias relative to the selected treatment's true mean (selective), and link them to regret from deploying a suboptimal treatment. This framework defines seven decision-relevant evaluation targets: mean bias, mean squared error, and confidence interval coverage for the global and selective winner's curse, and mean regret. We then show that methods that perform well on one target can perform poorly on others, so corrections should be matched to the manager's objective. Across simulations with varying effect sizes, multiple-arm settings, and data calibrated to an online A/B testing platform, no method dominates uniformly: the plug-in estimator performs best when treatment differences are large, cross-fitting performs best when treatments are similar, and resampling methods often achieve low mean squared error for moderate differences. We also introduce an adaptive empirical likelihood procedure that delivers asymptotically valid confidence intervals across settings without the tuning sensitivity of resampling-based methods.

Probabilistic prediction systems often aggregate probability estimates from multiple models into a single decision. A common assumption is that if each model is individually calibrated, the aggregate prediction will also be well calibrated. We show that this assumption fails in multi-agent settings: individually calibrated predictors can become collectively miscalibrated when their predictions interact strategically, in the game-theoretic sense of Brier-optimal local response, even without deliberate coordination. This phenomenon arises naturally when agents are independently trained on overlapping data. We prove that under Brier-score-based aggregation with positively correlated beliefs, each agent's individually optimal report systematically underestimates the positive-class probability, yielding a Price of Anarchy greater than one whenever Cov(b_i, b_j) > 0. In a canonical setting (n = 5 agents, pairwise correlation = 0.5, base rate = 0.3), the empirically measured PoA in false-negative rate reaches 7.25x. In contrast, VCG-based aggregation aligns incentives by rewarding marginal contribution, achieving dominant-strategy incentive compatibility and near-optimal performance. Experiments on three real-world datasets (NSL-KDD, UNSW-NB15, Credit Card Fraud) show that VCG provides strong robustness while maintaining comparable accuracy. It performs particularly well in data-sparse and adversarial settings, and adaptive weighting further improves performance under distribution shift.

We propose non-parametric estimators for the average run length (ARL) and average detection delay (ADD) in quickest changepoint detection (QCD) under finite and irregular sequence lengths. Although ARL and ADD are widely used as optimality criteria in theoretical and simulation studies, their application to real-world datasets is hindered by limited and irregular sequence lengths. To address this issue, we propose non-parametric estimators for the ARL and ADD, termed KM-ARL and KM-ADD, by drawing an analogy between QCD and survival analysis to model detection probabilities under sequence truncation. We derive estimation bias bounds and prove that they are asymptotically unbiased unless extrapolation is required. Experiments on simulated and real-world datasets demonstrate their practical utility, enhancing robustness against limited and irregular sequence lengths, improving interpretability, and facilitating empirical, intuitive model selection. Our Python code is provided at https://github.com/TaikiMiyagawa/Kaplan-Meier-Average-Run-Length, offering ready-to-use implementations for practitioners.

In this paper we consider the conditional stochastic optimization (CSO) problem. This consists of optimizing a function which can be written as the expectation of a function which is itself a function of a conditional expectation, i.e.~of the type $F(ξ) := \mathbb{E}\left[f\left(Z,\mathbb{E}[g(Z,X,ξ)|Z]\right)\right]$, where precise definitions are given in the main text. We address a particular class of CSO problems where the joint law of the random variables $X,Z$ cannot be exactly sampled; this case has been addressed in Goda & Kitade (2023). We introduce a method that combines Markovian stochastic approximation with unbiased approximation methods which allows one to find the optimizer of $F(ξ)$ in the context of interest. We illustrate our methodology on two examples associated to parameter estimation with model averaging and portfolio selection associated to high-dimensional full factor multivariate stochastic volatility models.

Scientific inference is often undermined by the vast but rarely explored "multiverse" of defensible modelling choices, which can generate results as variable as the phenomena under study. We introduce RobustiPy, an open-source Python library that systematizes multiverse analysis and model-uncertainty quantification at scale. RobustiPy unifies bootstrap-based inference, combinatorial specification search, model selection and averaging, joint-inference routines, and explainable AI methods within a modular, reproducible framework. Beyond exhaustive specification curves, it supports rigorous out-of-sample validation and quantifies the marginal contribution of each covariate. We demonstrate its utility across five simulation designs and ten empirical case studies spanning economics, sociology, psychology, and medicine, including a re-analysis of widely cited findings with documented discrepancies. Benchmarking on ~672 million simulated regressions shows that RobustiPy delivers state-of-the-art computational efficiency while expanding transparency in empirical research. By standardizing and accelerating robustness analysis, RobustiPy transforms how researchers interrogate sensitivity across the analytical multiverse, offering a practical foundation for more reproducible and interpretable computational science.