We propose a mixture of location-scale skewed-$t$ distributions to fit bimodal, skewed and heavy-tailed data. In particular, the mixture is based on the skewed-$t$ distribution by Fernández and Steel (1998), so that the model-building procedure can be easily extended to mixtures of other symmetric distributions. After studying the properties of the mixture, we develop a maximum likelihood estimation approach via the EM algorithm and a likelihood ratio test of the null hypothesis of no skewness in any given component. A simulation-based comparison to a recently proposed mixture of g-and-h distributions suggests that the performance of the proposed model is excellent, in terms of both estimation precision in well-specified setups and modeling capability in mis-specified frameworks. Fitting the model to the Standard & Poor's 500 distortion allows us to confirm the bimodality of its distribution, with the implication that the US stock market has historically been in bearish or bullish conditions, rather than near its fundamental value.

Uncertain data without frequency stability often arises in experimental design. Classical fixed-effects models can only analyze precise experimental data. Based on an uncertain measure, this paper establishes uncertain fixed-effect models for Latin-square designs. First, we propose three methods with uncertainty to estimate the treatment and blocked effects and construct their confidence intervals. Then, uncertain homogeneity and common tests are conducted to assess the significance of treatment effects. In the numerical simulations, the three estimation methods are compared based on bias, mean squared error, mean absolute error, overall standard deviation, coverage probability, and average interval length. Several examples are given to illustrate the process of estimation and hypothesis. Finally, the uncertain fixed-effects model is applied to real education data, demonstrating its practical value.

This paper investigates the problem of comparing the location parameters of two shifted exponential models through a novel double sequential sampling framework. The proposed hypothesis testing procedure is developed by controlling the type I error probability at a preassigned level while minimizing a loss function that incorporates both the type II error probability and the associated sampling cost. The corresponding optimal fixed-sample-size expressions are shown to depend on unknown scale parameters, rendering the desired testing accuracies unattainable in practice under fixed-sample designs. To overcome this difficulty, a double sequential sampling procedure is proposed to test the difference between location parameters when the scale parameters are unknown and unequal. The proposed methodology is shown to possess desirable asymptotic properties, including first-order efficiency, second-order efficiency, and second-order risk efficiency. Extensive simulation studies and a real-data application that involves heavy precipitation episodes at meteorological stations demonstrate the practical effectiveness and applicability of the proposed procedure.

Large-scale hypothesis testing supports probability claims about individual hypotheses, as in empirical Bayes methods for estimating local false discovery rates. We study how such claims can be interpreted as approximately calibrated forecasts of the null hypothesis, yielding interpretable error probabilities even under model misspecification. Our approach draws conceptual inspiration from probabilistic forecasting but addresses a different challenge: unlike forecasting, where labels are eventually observed, in multiple testing the ground truth is never revealed, so calibration must be assessed stochastically and established indirectly. We address this challenge by constructing a set of pseudo-labels, derived from the spacings of ordered $p$-values, which have the local false discovery rate as their regression target. Our construction unlocks existing tools for assessing and performing post-hoc calibration in multiple testing. Notably, we find on a large-scale empirical survey of published psychology and neuroscience literature that the $q$-value, a popular error measure based on the false discovery rate, can be severely miscalibrated.

Data encountered in practice are frequently contaminated by additive noise, and wavelet shrinkage remains a fundamental tool for recovering underlying signals in nonparametric estimation. Classical procedures such as hard and soft thresholding decide whether to retain a wavelet coefficient almost entirely from its magnitude. Although effective in many settings, these rules can be too rigid for coefficients whose magnitudes fall in an intermediate region where the distinction between signal and noise is uncertain. We propose MLShrink, a two-threshold wavelet denoising procedure that combines wavelet shrinkage with machine learning. Coefficients below a lower threshold are discarded, coefficients above an upper threshold are retained, and coefficients in the intermediate band are classified using local wavelet-domain features. In this way, MLShrink preserves the simplicity of classical thresholding away from the decision boundary while allowing data-adaptive decisions for ambiguous coefficients. The paper also develops a theoretical framework tailored to this architecture. We show that MLShrink is a nonexpansive support-selection rule, derive an oracle-based risk decomposition showing that excess denoising risk is determined by classification errors on the undecided band, and establish an oracle-consistency result under suitable assumptions on classifier performance. Simulation experiments on standard benchmark signals indicate that MLShrink is competitive with several established wavelet shrinkage methods and is especially effective for signals with irregular, edge-rich, or non-smooth structure. These findings suggest that learned decisions on the intermediate threshold band provide a useful and interpretable connection between classical wavelet denoising and modern statistical learning.

We introduce Symmetric CDF Oriented Probability Enhanced (SCOPE) shrinkage, a unified family of sign-preserving shrinkage rules constructed from centered cumulative distribution functions of symmetric unimodal distributions. The proposed framework generates a broad class of attenuation profiles that interpolate between strong local shrinkage near zero and asymptotically unbiased behavior in the tails. A general formulation is developed that separates scale and shape effects through two interpretable parameters, allowing effective threshold location and transition sharpness to be controlled independently. Under explicit regularity assumptions, structural properties of SCOPE shrinkage are established, including oddness, monotonicity, continuity, contractivity, and a mixture representation that connects the rules to softened thresholding operators. A Bayesian and penalized likelihood interpretation is also developed: SCOPE rules admit even penalty representations that are nondecreasing in coefficient magnitude, and suitable subclasses arise as exact maximum a posteriori estimators under proper symmetric unimodal priors. Representative examples based on logistic, uniform, and Cauchy distributions illustrate how probabilistic shape governs shrinkage behavior. Data driven parameter selection for smooth subclasses is discussed via Stein-type unbiased risk estimation. Oracle calibrated simulation studies on standard Donoho-Johnstone test functions show that SCOPE shrinkage performs competitively with several established wavelet denoising methods, while retaining a high degree of interpretability and structural flexibility. The results highlight centered distribution functions as a natural and versatile design principle for shrinkage in wavelet denoising and related estimation problems.

Active feature acquisition (AFA) sequentially selects which features to observe to reach a classification or ranking decision. Its central limitation is reliance on large amount of labeled data to fit probabilistic models guiding acquisition. Large language models (LLMs) supply unsupervised domain knowledge, but are poor sequential planners. Asking one to both know and decide conflates capabilities best kept separate. Here, we develop a framework for zero-shot AFA through disciplined elicitation: asking the LLM only for what it can be trusted to return, the unary deviations and pairwise co-variations that are the sufficient statistics of a Markov random field (MRF). We apply our framework to two settings: binary classification and top-$k$ identification. In practice, the LLM reliably returns only discriminative statistics, what distinguishes the classes rather than each class in isolation, which precludes classical AFA. We apply a maximum-entropy closure that resolves this gauge ambiguity. We evaluate on a cohort of Inflammatory Bowel Disease (IBD) patients, an active clinical setting where diagnostic ambiguity and patient heterogeneity obstruct stable treatment strategies. Our framework outperforms the LLM both on real labels and on its own extracted beliefs. Where it matters most, on the hardest patients, our top-$k$ acquisition policy markedly outperforms all existing methods.

Factor models are popular approaches for analyzing high-dimensional data to extract low-rank signals and estimate covariances. They decompose the covariance matrix as the sum of low-rank and diagonal components. A key issue is how to choose the latent dimension $k$, which is particularly challenging when the factor model only holds approximately and in low signal-to-noise scenarios. Bayesian overfitted factor models specify an upper bound on $k$ and rely on structured shrinkage priors to effectively remove extra components. Such approaches are popular and effective, but computationally expensive. We propose a much faster \texttt{EigenBayes} approach that provides valid uncertainty quantification, based on spectral estimation of latent factors and adaptive empirical Bayes calibration of key hyperparameters. The resulting posterior distribution factorizes across outcomes and is analytically tractable, bypassing Markov chain Monte Carlo. We show that \texttt{EigenBayes} adapts to the signal-to-noise ratio of each outcome and latent dimension, while shrinking superfluous latent components to zero. We establish favorable asymptotic properties and demonstrate strong empirical performance in numerical experiments and a genomics application, where EigenBayes outperforms state-of-the-art alternatives.

Motivated by the privacy, sensitivity and sharing limitations of health data, we present a comprehensive pipeline for inference of Bayesian mixture models within a federated learning setting, i.e. when data cannot be fully shared or pooled across compute nodes. We adopt a Consensus Monte Carlo (CMC) approach, in which an MCMC algorithm is run independently within each data silo to estimate local posterior distributions, which are then aggregated to approximate the posterior over the full data. The variational CMC approach of Rabinovich, Angelino and Jordan (2015) [1] frames the aggregation step as a variational inference problem, but their application to mixtures assumes the number of clusters and key mixture parameters to be known. Our main methodological contributions are: (i) an extension of variational CMC to over-fitted Bayesian mixture models that infer the number of clusters and all model parameters, without requiring conjugacy; (ii) novel cluster-matching algorithms suitable for cross-silo settings in which not every cluster appears in each local dataset; (iii) a number of inference strategies for the aggregation step, matched to different federated learning constraints; and (iv) guidelines for choosing among these in practice. A comprehensive simulation study validates the framework and allows us to compare to state-of-the-art federated learning alternatives. Notably, we show that when the composition of local datasets reflects the underlying clustering structure in the data, our approach can recover small clusters with greater accuracy than standard MCMC applied to the pooled data. We illustrate the framework on large-scale electronic health record data, identifying multi-morbidity patterns in a British geriatric population.

We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$--exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le β\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

We propose a Bayesian nonparametric approach using a truncated Enriched Dirichlet Process mixture (EDPM) model to estimate natural direct (NDE) and indirect (NIE) effects in causal mediation analyses in the presence of post-treatment confounders. We introduce an efficient cluster reallocation Metropolis-Hasting algorithm to improve mixing in the blocked Gibbs sampler. We implement a one-step posterior correction based on the efficient influence function for our setting. This post-processing step solves a critical problem in Bayesian nonparametrics: how to obtain reliable estimates and posteriors for a specific causal estimand of interest (the NDE and NIE) with excellent frequentist properties, such as correct coverage, from a model designed for complex joint distributions. We conduct simulation studies to assess our method's performance and apply it to evaluate causal mediation effects in a weight management clinical trial.

The g-formula is a foundational tool for identifying causal effects in observational data. This tool is based on the law of iterated expectation, a key mathematical identity in statistics. However, the notation with which the law of iterated expectation and the g-formula is expressed can be opaque to those with little background in statistics. We provide a primer introducing the law of iterated expectation, the integration notation used to express it, and its role for causal effect identification via the g-formula. Under the assumptions of causal consistency, positivity, and conditional exchangeability, the law of iterated expectation can be rewritten as a causal standardization formula (the g-formula) in two nonparametrically equivalent forms: a non-iterative conditional expectation (NICE) form involving a single weighted average of conditional outcome means, and an iterative conditional expectation (ICE) form involving nested expectations. We illustrate both forms using three progressively complex numerical examples: a time-fixed example with a single binary confounder, a time-fixed example with discrete and continuous confounders, and a time-varying example with two timepoints. We provide clarity on what the law of iterated expectation is, how it is related to the g-formula, and how to gain intuition of its mathematical formulations in actual data examples that can be generalized to a range of settings.

In offline Reinforcement Learning, immediate rewards in logged batch data are often unobserved due to sparse or irregular record-keeping, or censored beyond certain reward values. This issue arises in practical settings, including health care and marketing. We investigate off-policy evaluation (OPE) in finite-horizon Markov decision processes when rewards are missing not at random (MNAR), which breaks ignorability and induces selection bias even after conditioning on states and actions. To address this, we formalize a reward-dependent propensity model and use future states as shadow variables to identify the full-data conditional mean reward. We further introduce a bridge function that recovers the conditional mean reward without explicitly modeling the MNAR mechanism, and estimate it via a min-max procedure to avoid double sampling. Building upon these identification results, we propose an Fitted-Q-Evaluation-style estimator that propagates the recovered rewards while allowing target policies to depend on past missingness indicators. Finally, we establish consistency and finite-sample error bounds for our OPE estimator, and show through experiments the strong performance of our method compared to existing methods on simulated and MIMIC-III Sepsis data.

We study causal effect estimation with compositional treatments, where the exposure lies on a simplex and the estimand is defined over compositions rather than scalar or binary values. By considering a projection of the average potential outcome onto the treatment space, a kernel-based covariate functional balancing approach is adopted for weight construction. The weights are obtained by directly minimizing a worst-case balancing error over a reproducing kernel Hilbert space (RKHS) defined on the joint space of treatments and covariates, instead of being estimated under a treatment assignment model. Building on these weights, an augmented weighted estimator (AWE) is proposed, where the outcome function is estimated via kernel ridge regression and combined with a marginal augmentation over the covariate distribution. Despite the complex structure of the resulting objective, a finite-dimensional convex optimization problem is formulated via a representer theorem and a low-rank approximation. The proposed estimator achieves $\sqrt{n}$-consistency without requiring consistent estimation or smoothness of the weights. An asymptotic normality result is established around a sample-specific target. Empirical performance is demonstrated through simulation studies and a real data application.

Organizations and researchers show increasing interest in using large language models (LLMs) in place of human participants in A/B tests, in the hope of experimenting faster and at lower cost. We study when a treatment effect estimated on LLM outcomes can recover the effect that would have been measured on the human population of interest. Distributional equivalence between LLM and human outcomes would make any standard estimator valid but is unrealistic. We therefore develop a statistical framework that adapts surrogate endpoint theory to LLMs, showing that calibrating LLM outcomes to human outcomes identifies the average treatment effect under surrogacy and comparability conditions that are jointly weaker than distributional equivalence. We present a falsification test for surrogacy and a bound on the worst-case bias from limited overlap between the LLM and human samples. We further show that the stochasticity inherent to LLMs can weaken surrogacy for identification while also introducing bias and variance during estimation, but that using an average over multiple LLM draws per unit as the surrogate mitigates these issues. Simulations validate the results, and an empirical application to A/B tests on Upworthy headlines shows that raw LLM predictions recover only 39\% of the human treatment effect while nonparametric calibration closes the gap. A central takeaway is that A/B testing on LLMs yields correct results only by assumption, whereas A/B testing on humans is correct by design, and that the required assumptions are hardest to justify precisely where A/B testing on LLMs promises the greatest benefit. We discuss the role of LLM choice, prompting, and temperature as design variables, the compounded challenge posed by long-term outcomes, and how to size human pilot studies for validation.

Variable selection methods that control the false discovery rate often lose power when predictors exhibit complex dependence structures. We previously showed that selecting hierarchically clustered groups of predictors can mitigate this issue while maintaining false discovery rate control. When correlations are less structured, however, overlapping predictor sets may be more effective. We introduce a generalized false discovery rate for hypotheses defined on sets of predictors and propose a hypergraph-based selection method. This approach achieves higher power across diverse settings while preserving rigorous false discovery rate control.

We introduce a novel active-learning framework for failure probability estimation in structural reliability analysis that integrates Active Kriging Monte Carlo simulation with conformal prediction. The proposed approach employs an adaptive cross-conformal strategy specifically designed for small-sample settings and kriging surrogate models using the J+GP conformal estimator. Unlike standard AK-MCS methods, the proposed framework provides distribution-free guarantees on prediction errors, leading to more reliable classification of samples near the limit-state surface. This improved uncertainty quantification enhances both the accuracy and robustness of failure probability estimates, especially for rare-event regimes where such efficiency is crucial. Reproducible numerical results illustrate the effectiveness of the method and also compare it to classical approaches on well-established benchmarks.

Subset selection problems as MIPs (Mixed Integer Programs) are NP-hard. For large scale problems, it is infeasible to find global optimal solutions in a reasonable time and good-quality incumbent solutions are sought after with MIP solvers in practice. This paper proposes DASH (Decreasing Active Set Hierarchy) -- a dimensionality reduction method that improves the MIP solver performance for a subclass of best subset selection problems that can be formulated as MIQPs (Mixed Integer Quadratic Programs). We develop and evaluate the performance of DASH in the subset portfolio selection problem with comparison to Gurobi, a commercial MIP solver. In addition to the problem size, the difficulty of a problem is related to the condition number of the covariance matrix and the box constraint on portfolio weights. An extensive set of numerical experiments with varying problem configurations shows that DASH offers consistent and significant improvement of incumbent solutions when the problem is difficult to solve by Gurobi. In particular, the magnitude and duration of improvement by DASH scale with the difficulty of the problem.

Despite the theoretical appeal of their non-reversibility, to date, no Piecewise Deterministic Markov Process (PDMP) samplers have been developed that scale better than $\mathcal{O}(\sqrt{d})$ in computational complexity with respect to the target dimension $d$. We prove that this is a fundamental limitation by establishing an $Ω(\sqrt{d})$ lower bound on the algorithmic complexity of PDMP samplers in a standard setup. By relaxing the assumption that the target density must remain invariant at all continuous times, we then demonstrate how to bypass this barrier. Specifically, we introduce a novel PDMP sampling scheme and show that it achieves an empirical complexity of $\mathcal{O}(d^α)$, where $α\in [0.2, 0.3]$ for Gaussian-tailed targets. In addition, this PDMP scheme is locally adaptive in both trajectory length and distance between velocity updates.

Standard MCMC diagnostics ($\hat{R}$, effective sample size, divergence counts) detect whether a chain has mixed, but not why it has not. We ask whether the centring/non-centring obstruction in hierarchical models has a geometric cause beyond the metric. The joint parameter space is a fiber bundle (hyperparameters the base, group-level parameters the fibers), and the Fisher information metric induces an Ehresmann connection $A = -G_{FF}^{-1}G_{BF}$; the natural hypothesis is that the obstruction is its curvature, felt by the sampler as holonomy. We prove this false. The connection is flat for any smooth hierarchical posterior, not only the Gaussian case, because its horizontal leaves are the level sets of the fiber score $\partial_α\log p$: there is no geometric obstruction above the metric. What remains is statistical, not geometric, and the flat connection identifies it as a single quantity: the conditional dependence of fiber on base, governed per group by the prior fraction $π_j$, the classical pooling factor. From it the framework recovers the established picture, that prior-dominated groups mix slowly and that the optimal per-group non-centring weight follows in closed form, and a simulation study separates this base-fiber coupling from the funnel, a distinct base-space pathology, by their opposite dependence on the hierarchical variance. A direct attribution test confirms that NUTS does not transport the fiber: the chain-level footprint is excess conditional autocorrelation in prior-dominated groups, exactly as $π_j$ predicts. Genuine, even rotational, curvature does appear, but only for connections built from a sampler's working metric (a fixed mass matrix), where holonomy re-enters as an algorithmic rather than geometric phenomenon. The prior-fraction diagnostic is distributed as the R package fibr, with the geometric methods as accompanying reproduction code.

Identification conditions describe the computability of a target query or parameter of interest as a function of the type and amount of information available. In causal identification, this information is often expressed in the form of a causal graph, and data are observed or collected for some subset of variables in the graph. Target queries may be for a single effect alone or for a class of effects in a given model. The derivation of an identification algorithm then defines mathematically the process by which the desired causal effect(s) can be uniquely determined, theoretically, in expectation. Identifiability in expectation, or 'theoretical identifiability,' generally assumes asymptotic properties, infinite data, or other mathematically idealized conditions. In this paper, we explore a fundamental distinction between this theoretical, idealized notion of identifiability and a proposed alternative that is computation-bound. The framework we propose - 'computational identifiability' - is to instead define a finite computational search procedure for an empirical estimator. If this process finds an estimator empirically, within a desired error tolerance, then identifiability is satisfied, conditional on the specified assumptions of the search (i.e., a prior distribution over the parameters) and conditional on the search procedure itself. Through several experiments, we demonstrate how this framework allows us to answer fine-grained, practical identification questions, such as identification with small finite samples, with ambiguous graphical criteria, with mixed observational-interventional data, and across counterfactual data and estimands. Code is available at https://github.com/lbynum/metadentify.

Competing risks are commonly observed in engineering fields and can bring challenges to time-to-event data modeling when the application scenarios are complicated. Recently, deep neural networks have received great attention for prediction with competing risks, due to their flexibility and high learning capability. However, the complexity of neural network structure brings extra difficulty in hyperparameter tuning based on different data inputs. Additionally, when an engineered system has complex physical structures with multiple hierarchical levels, treating all structural levels as a single group of inputs may fail to capture critical information. To address the issues, we propose a Structured Segmented Hazard Deep Neural Network (SSH-Net) for failure time prediction under cause-specific competing risks framework. Our approach associates neural network structure with data structures, and allows different covariate groups to impact the failure prediction through separate sub-networks. The neural network is constructed based on a cause-specific competing risks model. The SSH-Net outputs cause-specific hazard functions, and utilizes the penalized log-likelihood as the loss function. The prediction accuracy of SSH-Net is validated through simulation studies by evaluating the Brier score, the area under receiver operating characteristic curves (AUC), and the root mean square error (RMSE) of the predicted cause-specific cumulative incident function. We further demonstrate the model's ability to predict failure time distribution functions using the Titan GPU failure time data.

Large language models (LLMs) are increasingly used as judges for open-ended generation, as large-scale human evaluation is often expensive and difficult to scale, yet their preferences remain imperfect proxies for human judgment. Existing auditing pipelines often assume that a reliable subset of examples or clean supervision signals are available beforehand, for example from human annotation, heuristic filtering, or the outputs of strong judges. In LLM evaluation, this assumption is fragile: the initial split may inherit judge bias, while human verification is typically too scarce to define stable groups at scale. We propose AURA, an adaptive uncertainty--aware refinement framework for auditing pairwise LLM--as--a--judge decisions under selected human verification. AURA iteratively learns a human-consistency signal, propagates reliable evidence, and prioritizes uncertain comparisons for human review. The key idea is to treat trust in a judge as a latent quantity that is progressively refined as evidence accumulates. We provide a compact formulation, a stable refinement procedure, and a comprehensive evaluation on both synthetic and real pairwise LLM-answer data.

We propose a scalable method for training prediction (machine learning) models in the predict-then-optimize paradigm, where model outputs serve as coefficients for a subsequent linear optimization task. Directly minimizing the empirical decision regret is intractable for linear programming and combinatorial optimization since the decision mapping is piecewise constant, and the gradients are zero almost everywhere. While existing methods address this by smoothing the differentiation process, they suffer from scalability issues, since a computationally expensive solver call is required for every gradient evaluation. To address this, we propose a decision-focused learning pipeline based on a measure transformation principle, which yields a new surrogate loss that is completely optimization-solver-free during training. We establish theoretical guarantees, including Fisher consistency and excess risk bounds. Empirically, our method achieves decision quality competitive with state-of-the-art methods while reducing training time by orders of magnitude.

Signed pairwise interaction scores fundamentally conflate uniqueness (U), redundancy (R), and synergy (S). We prove this on a minimal 3-way XOR structural causal model: faithful indices such as Shapley-Taylor return zero per pair, whereas projective indices such as Shapley Interaction spread the third-order effect into pair scalars that conflate the three mechanisms. We introduce Stochastic Hi-Fi, a post-hoc, retraining-free predictability decomposition that estimates per-feature U/R/S profiles by interventional masked inference. The estimator provides exact interventional semantics, finite-sample Monte Carlo bounds, strict variance reduction from coupled diamond sampling, and uniform finite-vocabulary convergence. Across tabular SCMs, Stochastic Hi-Fi recovers structure missed by scalar baselines (up to 411x larger interaction-magnitude recovery ratios). It also separates redundant and synergistic heads in the GPT-2 IOI circuit. On NIH ChestX-ray14, Stochastic Hi-Fi matches GradCAM on Pointing Game and improves substantially on Deletion AUC.

We study a multi-agent multi-armed bandit problem in the competitive setup with two-sided matching markets under a human centric decision making model. To capture human preferences, we use cumulative prospect theory (CPT) that weighs the actions of the agent in a nonlinear fashion using a ($α$-Hölder continuous) weight function. CPT has been widely used in behavioral economics and risk sensitive machine learning to emulate human preferences. We analyze the state-of-the-art learning algorithm with CPT weight distorted rewards and obtain a player optimal regret of $\mathcal{O}(K\log T \left(\frac{1}Δ\right)^{2/α})$, where $K$ denotes the number of arms, $T$ is the learning horizon, and $Δ$ represents (suitably defined) players' minimum preference gap. Noticing the dependence on $Δ$ to be sub-optimal, we further improve this regret by judiciously selecting the active set of arms during exploration, which removes the dependence on $K$ in the dominant term and achieves an improved (optimal) regret guarantees in the setting where the number of arms $K$ is significantly larger than the number of players $N$. In addition, we consider adversarial markets where the observed rewards of the agents may be corrupted. We propose and analyze algorithms for robust markets with CPT as risk sensitive measure in both settings where the total corruption budget is known and where it is unknown, and establish logarithmic player-optimal regret guarantees in both cases.

Preference-based post-training has become a central paradigm for aligning language models. A common data-collection strategy is to generate a small set of completions for each prompt and label the resulting comparison pairs. However, human preference labels are often much more expensive than generating additional completions, suggesting a different use of the same labeling budget: generate a larger pool of completions, but label only the most informative comparison pairs. This paper studies which pairs should be compared in preference-based post-training. We formulate comparison curation as a sampling-design problem and evaluate designs by the quality of the final policy under the preference-based post-training objective. We instantiate this framework for Direct Preference Optimization (DPO), analyzing how the choice of labeled pairs propagates through DPO training to downstream policy performance. Our main results provide matching upper and lower bounds on the post-training optimality gap of the DPO-trained policy. The bounds show that comparison selection affects downstream performance through a single design-dependent information matrix, which links label allocation to parameter estimation error and policy suboptimality. This yields an explicit optimization criterion for budgeted comparison curation and motivates practical sampling designs for selecting informative pairs from large generated completion pools. Experiments on synthetic settings and language-model post-training benchmarks show that the proposed designs consistently improve sample efficiency over common comparison-selection heuristics.

Pretrained transformers sit near singular minima of the loss, where the Fisher information metric degenerates along dead directions: directions in parameter space along which the directional Fisher vanishes. Locating such a direction normally needs a forward pass and an eigendecomposition of activations, or a sampling-based complexity estimate; none returns a direction computable from the network's parameters alone. We give one, for LayerNorm transformers. The inverse-scale direction $γ^{-1}/\|γ^{-1}\|$ of the LayerNorm affine is an exact algebraic kernel of the post-final-norm centred activation covariance, for any input distribution, and induces a corresponding dead direction in parameter space. It is read from the LN scale parameter alone, with no forward or backward pass and no eigensolve: the cheapest dead-direction read, specific to LayerNorm. We test it on $14$ pretrained transformers ($9$ LayerNorm, $5$ RMSNorm; $160$M-$35$B; language and vision objectives). At random initialisation the predicted direction matches the measured bottom singular direction (one forward pass, direct SVD) to four decimal places on $9/9$ LayerNorm models, and is correctly absent on $5/5$ RMSNorm models, which lack the mean-subtraction projector that creates it. On the trained checkpoint the covariance eigenvalue along this direction deepens by ${\sim}10^3\times$ and further dead directions open; the random-init-to-trained gap is a one-forward-pass, per-checkpoint readout of singular structure along the predicted coordinate. Two consequences follow in closed form: the residual stream's smallest singular value is preserved block-to-block on $13/14$ transformers measured on their own input distribution, the one exception (Gemma$4$-$31$B) a genuine dead direction the same read pinpoints; and the kernel direction's presence classifies a transformer's normalisation from the parameters alone.

A model is multicalibrated on a collection of group weights $G$ if it is calibrated -- i.e. unbiased even conditional on its prediction -- not just overall, but also after reweighting contexts by each $g \in G$. It is a useful property for many downstream applications and is a basic desideratum of trustworthy machine learning. Before this work, all predictors known to attain the minimax-optimal $\widetilde O(\varepsilon^{-3})$ sample complexity rate for $\varepsilon$-multicalibration were randomized, while deterministic predictors were known only with substantially worse sample complexity. Whether randomization is necessary for optimal sample complexity in multicalibration was explicitly asked by [CLNR26] and implicitly in several prior works. We resolve this open problem by giving a minimax-optimal multicalibration algorithm that outputs a deterministic predictor. We then generalize the algorithm to produce optimal deterministic predictors that satisfy outcome indistinguishability (OI) with respect to finite or finitely covered collections of tests. As an application, this also gives deterministic omnipredictors and panpredictors with optimal sample complexity, resolving open problems posed by [OKK25] and [BHHLZ25].

This paper considers stochastic linear contextual bandits (SLCB) with bounded reward noise. Existing works typically assume sub-Gaussian reward noise and bounded expected rewards, under which the optimal regret bound scales as $\tilde{O}(\sqrt{T})$ in terms of horizon $T$. However, in many applications, realized/observed rewards are also naturally bounded, implying bounded reward noise. Bounded noise is more informative than the sub-Gaussian condition but has not been leveraged explicitly in the SLCB literature. In this paper, we propose a novel algorithm SME-OFU by utilizing an uncertainty quantification method called set-membership estimation (SME) and applying the principle of optimism in the face of uncertainty (OFU). Our algorithm enjoys an improved regret bound $O(\log T)$. Notice that this does not contradict the existing optimal bound $\tilde{O}(\sqrt{T})$ for sub-Gaussian noise because bounded noise is a stronger condition. Finally, simulations show empirical improvements of SME-OFU over a benchmark algorithm designed for sub-Gaussian noise when the reward noise is bounded.

Recent work on first-order optimizers for empirical risk minimization (ERM) has suggested that smoothness of ERM loss functions in the training data, rather than in the optimization parameters, can be leveraged to improve the oracle complexity of gradient descent (GD) methods. In this paper, we propose an inexact gradient method, piecewise polynomial interpolation-based gradient descent (PPI-GD), which approximates the full gradient in each iteration by querying the first-order oracle at equidistant points in the data domain to construct polynomial interpolants of the resulting gradient samples over appropriately sized patches of the data domain. We analyze the oracle complexity of PPI-GD for strongly convex and non-convex loss functions when the data space dimension is bounded by a polylogarithmic function of the number of training samples, and find it to outperform several GD variants in key regimes when the loss function is sufficiently smooth. Furthermore, our analysis extends several techniques from the error analysis of bicubic spline interpolants to the setting of $d$-variate tensor product polynomial interpolants which may be of independent interest in interpolation analysis.

In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.

Deep learning has managed to evade numerous intuitions from classical statistics to achieve unprecedented performance on a number of real-world tasks. In this article, we investigate the key features and surprises of deep learning from a physics-informed perspective, taking care to point out and justify where possible the many choices inherent in constructing a deep learning model. In particular, we review the phenomenon of neural scaling laws and discuss their interplay with the constraints and inductive biases which may be present when applying machine learning to problems in physics.

Unanchored indirect treatment comparisons using single-arm studies or disconnected evidence are increasingly used in health technology assessment (HTA) when randomized evidence is unavailable. Existing methods, including matching-adjusted indirect comparison (MAIC) and simulated treatment comparison (STC), are generally limited to pairwise settings and typically estimate marginal effects in the comparator study population, which may differ from the decision-relevant population. We propose multilevel unanchored meta-regression (ML-UMR), a Bayesian regression framework for synthesizing individual patient data and aggregate data from fully disconnected evidence. ML-UMR extends multilevel network meta-regression (ML-NMR) to unanchored settings by jointly modeling individual- and aggregate-level data within a unified likelihood, enabling estimation of treatment-specific outcomes and both marginal and conditional effects across multiple treatments, studies, and target populations. ML-UMR distinguishes assumptions required to identify treatment effects from those required to transport results to target populations. As with all unanchored comparisons, valid inference relies on strong and often unverifiable assumptions, including conditional exchangeability, correct specification of the outcome model, and cross-treatment assumptions (e.g., shared prognostic factor assumption (SPFA)). ML-UMR does not lessen these requirements but makes them explicit within a unified framework and facilitates sensitivity analyses. In simulation studies, ML-UMR produced low bias and nominal coverage for comparator-population effects. Transportability to alternative populations depended critically on identifying assumptions: violations of SPFA led to bias under strong effect modification, whereas incorporating subgroup information restored near-unbiased estimation and nominal coverage.

In time-to-event analysis, the hazard ratio (HR) derived from the Cox proportional hazards (PH) model is the most commonly used and widely reported measure for assessing treatment effects. However, hazard ratios are non-collapsible due to their inherent conditioning on survival up to each time point. As a result, they are subject to built-in selection bias in the presence of unmeasured heterogeneity arising from omitted important covariates, even when these covariates are independent of the main exposure at baseline, as is the case in randomized controlled trials. This article aims to provide an overview of key findings from the literature on how unobserved heterogeneity, due to omitted covariates that affect the outcome, can bias the estimation of the treatment hazard ratio in standard proportional hazards models, even in randomized trials where treatment is assigned independently of such covariates. Through simulations, we evaluate the extent of bias in the semi-parametric Cox PH model and parametric PH model under various scenarios of unmeasured heterogeneity. We then compare these standard models to alternative approaches that either account for this issue or are considered robust to it. These alternatives include the hazard ratio estimated from frailty models, regression parameters from an Accelerated Failure Time (AFT) model, and survival differences between treatment groups estimated nonparametrically using Kaplan-Meier curves or based on a Cox model with time-dependent effect of the exposure. We illustrate the practical relevance of the explored alternatives through a real data application to a randomized controlled trial from the Radiation Therapy Oncology Group (RTOG 9202).

Recurrent events or competing risks regression models are often applied in the bio-medical setting and both can be considered as marginal models. In presence of right-censoring, such models need to be adjusted to give consistent estimators. When censoring is administrative, marginal regression models are particularly easy to estimate. However, when censoring is instead acting randomly, inverse probability of censoring weighting (IPCW) adjustments are typically considered to obtain parameter estimates. This technique relies on a censoring-weights adjustment via a correct censoring model, but for administrative censoring the adjustment is done correctly simply by modifying the risk-set. In practice for large central registries or some clinical trials, the administrative censoring time will be known for all subjects, but there will typically also be a proportion of subjects that are censored at random. In this work, we consider two frequently used regression approaches, the Ghosh-Lin model for recurrent events with terminal events and the Fine-Gray model for competing events. For these two settings, when both administrative and random censoring are present, we demonstrate how to obtain correct estimation by dealing with the combination of the two different types of censoring relying on a minimum of modeling assumptions.

Physical activity plays a vital role in human health. Its entire distribution differs among people. Commonly used summary measures cannot describe this distributional pattern. We present a distribution-based analytical approach to describe physical activity by modeling individual-level activity-intensity patterns through hazard functions derived from wrist-worn accelerometer data. We analyzed minute-level Monitor-Independent Movement Summary (MIMS) data of 4297 adults with seven continuous days of device wear from the 2011- 2012 National Health and Nutrition Examination Survey (NHANES). We derived a nonparametric activity-intensity hazard using a survival-based approach for each individual on a common intensity grid, treating both the hazard curves from MIMS and their log-transformed MIMS as functional objects. We used functional principal component analysis (FPCA) on both scales of MIMS to characterize dominant modes of variation in activity-intensity distributions. Group-wise mean hazard functions showed little difference at lower intensity levels, while we observed a substantial difference at higher intensity levels. Our results demonstrate that hazard-based functional representations for capturing differences in physical activity intensity distributions across individuals offer a flexible and interpretable way to characterize heterogeneity. This approach works better than mean-based summaries and supports principled comparisons of physical activity patterns across population subgroups.

Large scale genetic datasets often aggregate the total allele counts of distinct genetic markers. Inferring haplotype frequencies (i.e.\ the frequency of multimarker alleles) from these pooled data is a challenge. Previous spatio-temporal modelling in this context has been limited to 3 markers due to the computational cost. In this work, we propose a nearest neighbor Gaussian process (NNGP) model to improve scaling with the number of markers and observations. To infer the parameters of our model, we develop a novel sequential Monte Carlo squared algorithm, which uses particle Gibbs with ancestor sampling to mutate the NNGP function values. The latter has a linear cost in the number of observations and the number of NNGPs, and can be applied to a broad range of NNGP models. As a case study, we analyse genetic data relating to antimalarial drug resistance in Africa, and show our scaling results empirically on a 3 and 6 genetic marker dataset.

In the last years, an anomalously high spreading of West Nile virus (WNV) has been observed in Italy, with particularly high peaks of infections in southern Lazio, Campania and Veneto regions. The main disease vector for WNV is represented by Culex pipiens mosquitoes, which spread human infections through their bites. Here, we investigate WNV fever epidemic diffusion during summer season 2025 in Italy through a computational approach based on a quantum version of the Game of Life (GOL) cellular automaton model. Specifically, human dynamics evolves according to the GOL rules, while stochastic dynamics of disease vectors, i.e., mosquitoes, as well as their interaction with humans, simultaneously occur. We show that this model fits the curves of cumulative infected individuals with high accuracy, either at local and average-regional level, with only optimization of mosquito birth and removal rates parameters. Furthermore, leveraging model flexibility, we show that changes in model parameters values elucidate system response to environmental variations. For instance, we quantify, e.g., the impact of mosquito spreading containment measures or sudden mosquito increasing abundance due to climatic and ecological changes. Overall, we provide a general, quantitative description of WNV infection spreading in Italy which could represent a supportive tool to test different environmental scenarios and could be useful to devise strategies for decision makers to monitor disease vector dynamics and to control consequent virus diffusion.

Population-based biobanks, now established in many countries, offer opportunities for large-scale studies investigating the incidence of various diseases. Biobank data is typically collected from a study cohort recruited over a defined calendar period, with subjects entering the study at various ages falling between $R_L$ and $R_U$. This work focuses on biobank data that includes individuals in whom onset of the disease of interest occurred before recruitment, termed prevalent cases, along with individuals initially recruited as disease-free in whom disease onset occurred during the follow-up period. We propose a novel cumulative incidence function (CIF) estimator that goes beyond existing methods in that it incorporates all disease cases, both prevalent and incident, irrespective of their subsequent life course. In particular, the new method can handle situations involving diseases that can occur at young ages with long survival after disease onset. Asymptotic properties of the new method are established and a simulation study is presented examining the performance of the method. We illustrate the use of the method and highlight its advantages over existing methods with an application to cancer data from the UK biobank.

Two-sample IV is a popular estimation method when the outcome and treatment variables are available in different samples, whereas instruments are available in both samples. The standard estimator is two-sample two-stage least squares estimator, which is efficient under homoskedasticity and homogeneity of the samples. We develop a robust two-step procedure for efficient estimation under general heteroskedasticity and heterogeneity of the samples, and propose a related two-sample Hansen overidentification test. A key feature of our approach is that only summary statistics from the linear regressions of the reduced form and first-stage in the two samples are needed. These are the six objects of the estimated coefficient vectors, and the homoskedastic and heteroskedasticity robust estimated variance matrices. We further show that the first-stage F-statistic in the treatment sample can be used as a test for weak instruments in the standard way under homoskedasticity and homogeneity, with the relative bias here a proportional bias. We propose an extension of the effective F-statistic of Montiel-Olea and Pflueger (2013) for the heteroskedastic case, following the generalization in Windmeijer (2025). We illustrate the estimators and tests in an application studying the effect of education on voting behavior from Marshall (2019), with cluster robust inference.

The accurate calibration of interest rate models is central to market-consistent valuation and Economic Scenario Generators (ESGs). Traditional calibration methods for multi-factor models such as the G2++ model often rely on point estimates, neglecting the influence of specific market data and the quantification of estimation uncertainty. This paper develops a diagnostic framework embedding the calibration problem into non-linear regression theory. It shows that the common industry practice of minimizing the Root Mean Squared Relative Error (RMSRE) is equivalent to a Weighted Least Squares (WLS) problem. This equivalence yields the corresponding formulations for diagnostic tools, including the Weighted Hat Matrix for leverage analysis, Influence Functions for local sensitivity diagnostics, and the Functional Delta Method for local, boundary-respecting confidence intervals. The implementation uses an efficient Jacobian factorization that exploits the analytical tractability of At-The-Money (ATM) caps. The framework is applied to a dataset of Euro ATM caps covering the period 2016--2025. Our empirical analysis reveals a boundary-dominated leverage profile, repeated losses of effective dimensionality due to active parameter constraints, and a diagnostic regime shift in local parameter stability around the post-2022 market transition. The resulting message for actuarial model governance is that low RMSRE is not sufficient for calibration validation. We conclude by discussing the framework's applicability to general least-squares problems while highlighting the computational challenges for instruments lacking closed-form gradients, such as swaptions.

We study dynamic resource allocation in a multicore computing system with a fixed number of processing cores and a stream of {\it malleable} jobs. Each job may adjust its level of parallelism during execution, allowing adaptive redistribution of resources across concurrently active jobs. Jobs belong to one of two observable classes, each characterized by a distinct speed-up function with unknown parameters. The objective is to learn a core-allocation policy that minimizes the long-run mean number of jobs in the system, equivalently the mean response time in steady state. \noindent To address this uncertainty, we develop an iterative learning-and-control framework. The system alternates between estimating the unknown speed-up parameters from observed job completions and solving the associated Markov decision process (MDP) to update the allocation policy. Within each job class, cores are shared equally among active jobs; the fraction of capacity assigned to each class is obtained from the MDP formulation of \cite{berg2017}, evaluated at the current parameter estimates. We construct a maximum likelihood estimator based on state-dependent inter-departure times and prove its strong consistency under a fixed allocation policy. We further propose two learning algorithms that combine this estimation step with dynamic programming-based policy updates, and illustrate their through numerical experiments.

We reconcile privacy protection and rate-double-robust inference. The privacy of individuals is protected by a local privacy mechanism: injecting noise into their sensitive data, revealing only the noisy data for inference. Hence, privacy protection hinders inference. In contrast, the inference of a target parameter is rate-double-robust when the large-sample bias of an estimator of the parameter is characterised by a trade-off between the estimation errors of two other, nuisance, parameters. Hence, rate-double-robustness facilitates inference. Our starting point of reconciliation is a class of rate-double-robust target parameters indexed linearly by an infinite-dimensional and nonlinearly by a low-dimensional regression. Among others, this includes causal parameters. To infer these targets privately, we show how suitable privacy mechanisms transfer the semiparametric properties of the sensitive-data model to the private setting. Rate-double-robustness is transferred, enabling locally-private, unbiased and semiparametrically efficient inference of our target parameters. Finally, we transform general nonparametric nuisance estimators into private ones, which inherit convergence properties of their nonprivate counterparts. For parametric nuisance models, we develop a private method-of-moments estimator and its large-sample inference theory.

The statistical modeling of consensus in Delphi data faces a critical bottleneck: the high dimensionality of questionnaire items relative to the limited sample size of expert panels. This rank deficiency leads traditional latent variable models, such as Principal Component Analysis, to be structurally unstable and prone to overfitting. Addressing this methodological gap, this study proposes a transition from variable-centric covariance models to network-centric connectivity models. By mapping item correlations onto a weighted graph topology, we present a simulation-based benchmark that utilizes community detection algorithms to identify latent thematic structures, effectively addressing the spectral instability and rank deficiency typical of high-dimensional, low-sample-size regimes. The research systematically evaluates the robustness of topological approaches based on structural density, information flow, and spectral partitioning against synthetic datasets designed to replicate the pathological conditions of consensus data, including ordinal scales and systemic noise. The central methodological contribution lies in demonstrating that collinearity among expert judgments - traditionally treated as statistical redundancy to be regularized - can be effectively reinterpreted as a topological signal of cohesion. This framework provides researchers with a structured and automated procedure for dimensionality reduction, ensuring structural stability and psychometric consistency even in small-sample regimes where standard factor analysis breaks down.

Many real-world networks are incomplete, making link prediction a fundamental challenge in network science. To train parameters and evaluate algorithms, observed links are usually divided into three subsets, namely training, validation, and probe sets. This division implicitly involves two sampling processes: first-stage sampling yields the probe set and second-stage sampling obtains the variation set. To date, our understanding of how these two sampling processes affect algorithm performance remains quite limited. To address this issue, we propose a sampling scheme called $β$-sampling, where the sampling probability of a link is proportional to the product of the degrees of its two endpoints raised to the power of $β$. Experiments on 45 real-world networks reveal that the structural characteristics of missing links, as simulated via varying probe sets, substantially impact prediction accuracy. When missing links tend to connect high-degree nodes, such links can be predicted accurately with ease. Furthermore, even with a fixed probe set, second-stage sampling still exerts a significant influence on prediction accuracy. Notably, the optimal second-stage sampling strategy differs from \textit{random sampling} (which randomly selects links to form the validation set) and \textit{consistent sampling} (which guarantees that links in the validation and probe sets share identical structural characteristics).

Probabilistic weather forecasting is undergoing rapid transformation with artificial intelligence (AI). In traditional numerical weather prediction, computing power can limit how well ensemble forecasts approximate the unknown statistical distribution of future states. AI models facilitate larger ensembles and are trained with probabilistic considerations, ideally leading to better uncertainty quantification. Forecasts from these state-of-the-art models are often considered well-calibrated. However, here we show that the statistical coverage of such models, the ultimate measure of calibration, can struggle, especially on extreme events. To address this shortcoming, we employ conformal prediction, a class of statistical methods that mathematically guarantees coverage under no distributional assumptions, unlike previous post-processing techniques. We apply online conformal prediction to temperature and precipitation forecasts (including extremes) of three leading global weather models, GenCast, NeuralGCM, and AIFS-ENS, ensuring calibrated uncertainty at no expense to other probabilistic metrics. This post-processing method can be applied to any forecasting model.

A game of chess is a stream: a time-ordered sequence of moves, each carrying an engine evaluation, a measure of accuracy, a measure of position complexity, and a clock reading. We model a game as a multivariate path and apply the signature transform of rough-path theory to obtain a reparametrization-invariant, graded feature set that records the order and interaction of in-game events without a parametric likelihood. We show that a player's law of play is identifiable from the expected signature up to tree-like equivalence, construct a signature-kernel two-sample test on path space, and recast cheating detection as an anytime-valid sequential test: a signature conformance score becomes an e-process whose error is controlled for every sample size at once by Ville's inequality, with fluctuations calibrated on the moderate-deviation scale. The discriminating information lives in the signature's Levy areas, which measure whether accuracy rises precisely when positions become hard--the fingerprint of engine assistance that aggregate match-rate statistics discard. In a controlled study the test holds exact type-I control and detection power rises from negligible for subtle assistance to 0.98 for blatant assistance, with a median detection time matching the growth-rate prediction. Calibrated to Magnus Carlsen's documented elite accuracy, the monitor does not flag world-champion-level play; and we exhibit cheating strategies that leave every aggregate statistic, including the best-move-frequency z-score of the Regan system, unchanged yet are caught cleanly by the signature--making precise how an order-aware, anytime-valid test strengthens the prevailing approach to chess anti-cheating.

Sparse point observations are increasingly available for precipitation nowcasting, but it is unclear how much they improve dense radar-field forecasts. We partially address this question with a multimodal graph neural network nowcasting system over the Nordic radar domain. The model predicts rain rate every five minutes up to two hours ahead and is trained with different combinations of radar history, MEPS numerical weather prediction, Netatmo surface observations, MSG satellite channels, stochastic noise, and CRPS-based ensemble losses. The study is designed as an ablation of operationally relevant information sources and training objectives. We compare radar-only, NWP-informed, station-informed, satellite-informed, noise-augmented, and CRPS-based configurations using complementary diagnostics on the radar grid, at station locations, for rain onset, and through oracle, displacement, and amplitude scores. The results show that each source improves a different part of the forecast problem. MEPS stabilises radar-only extrapolation, Netatmo observations improve local station and onset diagnostics, and satellite predictors reduce some station-level biases but may activate rain too early when used deterministically. CRPS-based configurations provide the most consistent radar-grid gains, while the combined satellite and CRPS setup gives the best overall oracle/DAS score. These results do not support the conclusion that point observations are uninformative for nowcasting, but they show that local observational skill and spatially coherent radar-field skill are distinct targets. The practical implication is that sparse observations can provide useful local constraints, but their benefit for radar-like fields depends on the training loss, uncertainty representation, and how observation support is encoded in the model.

We propose a parameter-efficient speech enhancement framework, Quaternion Conformer GAN (QC-GAN), which combines a Quaternion Conformer generator with MetricGAN-based training. The Hamilton product encodes the magnitude and phase via structured weight sharing, reducing the number of layer parameters while preserving their interdependencies. A metric-learning discriminator was employed to maximize perceptual quality by optimizing the approximate perceptual evaluation scores. On the VoiceBank+DEMAND dataset, QC-GAN achieved a Perceptual Evaluation of Speech Quality (PESQ) score of 3.48 with only 0.89M parameters, delivering a performance comparable to state-of-the-art models at less than half their size. A 35K-parameter variant achieved a PESQ score of 3.23, surpassing conventional methods with significantly fewer parameters. Evaluation on the DNS-Challenge 3 dataset further confirmed generalization to real-world conditions.

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

We develop a partial differential equation formulation of the Behrens-Fisher problem for two independent normal samples with unknown and unequal variances. An orthogonal decomposition separates mean and residual components (corresponding to the centered within-sample variation left after removal of the mean directions) and recasts the studentized difference of sample means as a scale-invariant geometric constraint. This reduction transforms the distributional problem into the evaluation of spherical wedge probabilities, which are identified with harmonic measure and with the value at the origin of a Laplace-Dirichlet boundary value problem. From this framework, we derive exact finite-sample representations for the cumulative distribution function and the probability density function in terms of beta functions, with dependence only on the sample sizes and the variance ratio. These representations place the Behrens-Fisher law in a standard special-function form that is directly accessible in widely available commercial software -- including Microsoft Excel -- thereby facilitating distributional evaluation and quantile computation. We also obtain a Gegenbauer separation-of-variables expansion for the associated harmonic extension and its threshold derivative, with coefficients in closed Beta-Gamma form, and derive sharp tail expansions with explicit leading constants and higher-order corrections.

We develop indexed Bellman information complexity, a representation-level theory of interactive decision making centered on information indices and reference histories. The representation strips away problem-specific syntax and retains only the ingredients needed for dynamic programming and information accounting, thereby unifying the earlier framework of indexed algorithmic information ratios (AIR). On the upper-bound side, regret is controlled by Bellman supersolutions or potential identities whose gradient bracket is paid for by indexed information. Upper-confidence-bound (UCB), estimation-to-decision/decision-estimation-coefficient (E2D/DEC), and adaptive-minimax-sampling or exploration-by-optimization (AMS/EBO) methods appear as three relaxations of this same identity. On the lower-bound side, the posterior-reference trajectory supplies both the information telescope and the ghost quantile of small-regret trajectories. The resulting critical radius in the lower bound is an effective-dimension-scale quantity, as in Fano and local-prior-mass lower bounds, rather than the constant radius of a two-point Le Cam argument. The examples show that DEC is best viewed as a one-step relaxation of indexed Bellman information complexity, not as a universally tight conversion mechanism. We illustrate the framework through several applications, with particular emphasis on kernel bandits. In this setting, the active action marginal provides a concrete basis for comparing UCB, E2D, and AMS/EBO.

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

Machine learning is central to modern science, industry, and policy, yet its predictive power often comes at the cost of transparency: we rarely know which input features truly drive a model's predictions. Without such understanding, researchers cannot draw reliable conclusions, practitioners cannot ensure fairness or accountability, and policymakers cannot trust or govern model-based decisions. Existing tools for assessing feature influence are limited; most lack statistical guarantees, and many require costly retraining or surrogate modeling, making them impractical for large modern models. We introduce AICO, a broadly applicable framework that turns model interpretability into an efficient statistical exercise. AICO tests whether each feature genuinely improves predictive performance by masking its information and measuring the resulting change. The method provides exact, finite-sample feature p-values and confidence intervals for feature importance through a simple, non-asymptotic hypothesis testing procedure. It requires no retraining, surrogate modeling, or distributional assumptions, making it feasible for large-scale algorithms. In both controlled experiments and real applications, from credit scoring to mortgage-behavior prediction, AICO reliably identifies the variables that drive model behavior, providing a scalable and statistically principled path toward transparent and trustworthy machine learning.

The digitalization of credit scoring has become essential for financial institutions and commercial banks, especially in the era of digital transformation. Machine learning techniques are commonly used to evaluate customers' creditworthiness. However, the predicted outcomes of machine learning models can be biased toward protected attributes, such as race or gender. Numerous fairness-aware machine learning models and fairness measures have been proposed. Nevertheless, their performance in the context of credit scoring has not been thoroughly investigated. In this paper, we present a comprehensive experimental study of fairness-aware machine learning in credit scoring. The study explores key aspects of credit scoring, including financial datasets, predictive models, and fairness measures. We also provide a detailed evaluation of fairness-aware predictive models and fairness measures on widely used financial datasets. The experimental results show that fairness-aware models achieve a better balance between predictive accuracy and fairness compared to traditional classification models.

Infants acquire language with generalization from minimal experience, whereas large language models require billions of training tokens. What underlies efficient development in humans? We investigated this problem through experiments wherein robotic agents learn to perform actions associated with imperative sentences (e.g., push red cube) via curiosity-driven self-exploration. Our approach amortizes active inference using Q-learning, enabling intrinsically motivated developmental learning. The simulations reveal key findings corresponding to observations in developmental psychology. i) Generalization improves drastically as the scale of compositional elements increases. ii) Curiosity-driven exploration enables faster learning. iii) Rote pairing of sentences and actions precedes compositional generalization. iv) Exception-handling induces U-shaped developmental performance, a pattern like representational redescription in child language learning. These results suggest that curiosity-driven active inference accounts for how intrinsically motivated sensorimotor-linguistic learning supports scalable compositional generalization and exception handling in humans and artificial agents.

We propose a new approach for the modeling large datasets of nonstationary spatial processes that combines a latent low rank process and a sparse covariance model. The low rank component coefficients are endowed with a flexible graphical Gaussian Markov random field model. The utilization of a low rank and compactly-supported covariance structure combines the full-scale approximation and the basis graphical lasso; we term this new approach the full-scale basis graphical lasso (FSBGL). Estimation employs a graphical lasso-penalized likelihood, which is optimized using a difference-of-convex scheme. We illustrate the proposed approach on synthetic fields as well as with a challenging high-resolution simulation dataset of the thermosphere. In a comparison against state-of-the-art spatial models, the FSBGL performs better at capturing salient features of the thermospheric temperature fields, even with limited available training data.

In this paper, we develop the mathematical framework for filtering problems arising from biophysical applications where data is collected from confocal laser scanning microscopy recordings of the space-time evolution of intracellular wave dynamics of biophysical quantities. In these applications, signals are described by stochastic partial differential equations (SPDEs) and observations can be modelled as functionals of marked point processes whose intensities depend on the underlying signal. We derive both the unnormalized and normalized filtering equations for these systems, demonstrate the asymptotic consistency and approximations of finite dimensional observation schemes respectively partial observations. Our theoretical results are validated through extensive simulations using synthetic and real data. These findings contribute to a deeper understanding of filtering with point process observations and provide a robust framework for future research in this area.

We develop a statistical toolbox for a quantitative model evaluation of stochastic reaction-diffusion systems modeling space-time evolution of biophysical quantities on the intracellular level. Starting from space-time data $X_N(t,x)$, as, e.g., provided in fluorescence microscopy recordings, we discuss basic modelling principles for conditional mean trend and fluctuations in the class of stochastic reaction-diffusion systems, and subsequently develop statistical inference methods for parameter estimation. With a view towards application to real data, we discuss estimation errors and confidence intervals, in particular in dependence of spatial resolution of measurements, and investigate the impact of misspecified reaction terms and noise coefficients. We also briefly touch implementation issues of the statistical estimators. As a proof of concept we apply our toolbox to the statistical inference on intracellular actin concentration in the social amoeba Dictyostelium discoideum.

Sparse methods (e.g., Best Subset Selection, Elastic Net) are the standard approach for obtaining interpretable models, but they can suffer from high variance and vulnerability to spurious correlations. Alternatively, algorithmic ensembles (e.g., Random Forests, Gradient Boosting) achieve high prediction accuracy but yield uninterpretable black boxes driven by randomization or sequential residual fitting. In recent years, a unifying paradigm has emerged: Objective-Driven Ensembles. By generalizing best subset selection into a joint mathematical optimization problem, this approach generates interpretable ensembles by optimally splitting predictors across a small number of diverse models. In this paper, we synthesize this growing body of literature and illustrate the statistical principles driving its empirical success. Specifically, we utilize finite-sample bounds to demonstrate how penalizing predictor overlap controls ensemble covariance and provides a mathematical hedge against spurious correlations. We evaluate these mechanics using an exact combinatorial oracle, and review how recent computational approximations have successfully scaled this framework to a variety of domains, including high-dimensional data, classification tasks, and settings with casewise or cellwise contamination, achieving machine-learning-level accuracy while retaining the interpretability of sparse models.

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $\mathbb{R}$. We then introduce a variant of the classic mapper graph of Singh et al.~(2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $(\mathbb{X}, f)$ when it is applied to points randomly sampled from a probability density function concentrated on $(\mathbb{X}, f)$. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of $(\mathbb{X}, f)$, a constructible $\mathbb{R}$-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $(\mathbb{X},f)$ to the mapper of a super-level set of a probability density function concentrated on $(\mathbb{X}, f)$. Finally, building on the approach of Bobrowski et al.~(2017), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.

We introduce the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information, 2) statistical models are fitted to distributions over these topological features, and 3) the target type classification methods of Wigren and Bar Shalom et al. are employed to exploit the resulting likelihoods for topological features inside of the tracking procedure. To demonstrate the efficacy of our approach, we test our procedure on synthetic vehicular data generated by the Simulation of Urban Mobility package.