Papers

In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>\alpha_0$, where $\alpha_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.

We investigate the ability of transformers to perform in-context reinforcement learning (ICRL), where a model must infer and execute learning algorithms from trajectory data without parameter updates. We show that a linear self-attention transformer block can provably implement policy-improvement methods, including semi-gradient SARSA and actor-critic, via explicit parameter constructions. Beyond existence, we design a teacher-mimicking training procedure, analyze its gradient-flow dynamics, and establish the first convergence guarantee in the ICRL literature: under suitable richness conditions on the training MDP distribution, gradient flow converges locally and exponentially to an optimal parameter manifold corresponding to the desired RL update. Empirically, training transformers on randomly generated tabular MDPs confirms these predictions: the learned models recover the parameter structure of our explicit constructions and, when deployed on unseen MDPs, deliver strong in-context control performance. Together, these results illuminate how transformer architectures internalize and execute classical reinforcement learning algorithms in context, bridging mechanistic understanding and training dynamics in ICRL.

The role of AI-generated synthetic data has recently been expanded to support realistic Monte Carlo simulations. However, guidance is limited on generating data with multilevel structures and designing simulations based on such data. This study proposes a general framework for AI-based simulation studies to evaluate the predictive performance and parameter recovery of quantitative methods, specifically using multilevel data commonly observed in the social sciences. Our proposed six-stage workflow consists of (i) specifying a method and real data, (ii) training Generative AI with real data, (iii) assessing synthetic data quality, (iv) designing and conducting simulations, (v) evaluating method performance, and (vi) checking robustness. To enhance fidelity in multilevel data generation, we also introduce targeted modifications to diffusion models and Generative Adversarial Networks (GANs). Furthermore, we develop a systematic quality evaluation framework that assesses both within-table and between-table fidelity, and discuss how AI-based simulation designs should differ depending on whether the simulation's objective is predictive performance or parameter recovery. Finally, using empirical multilevel data and multilevel modeling methods, we demonstrate the utility of the proposed AI-based simulation framework. This approach leads to more accurate and honest evaluations of quantitative methods in the real world, unlike traditional simulation studies based on arbitrary simulated scenarios.

The discrete Pareto (or Zeta, Zipf) distribution, arises naturally in modeling rank-frequency data across diverse fields such as linguistics, demography, biology, and computer science. Despite its widespread applicability, goodness-of-fit testing for the discrete Pareto distribution remains underdeveloped, particularly in the presence of heavy tails and infinite support. This article introduces a novel goodness-of-fit test based on a new Stein-type characterization of the discrete Pareto distribution, formulated using its probability generating function. The proposed method is applicable even when the shape parameter is unknown and avoids binning or smoothing techniques. We study the asymptotic properties of the test and assess its empirical size and power through extensive simulation experiments. The results show that the proposed test either outperforms or matches the performance of existing method across various alternatives. Applications to real datasets are provided to demonstrate its practical relevance and robustness.

Gaussian process (GP) marginal likelihood scores and kernel conditional independence tests are theoretically appealing for nonlinear causal discovery but computationally prohibitive at scale. We present three complementary RFF-based methods forming a practical toolkit for score-based, constraint-based, and hybrid causal discovery. The Fourier Feature Marginal Likelihood (FFML) score approximates the exact GP marginal likelihood by replacing the $n x n$ kernel Gram matrix with a finite-dimensional feature representation, reducing cost to $O(nm^2 + m^3)$ while retaining the probabilistic interpretation and automatic complexity penalty of the exact score. FFML extends to mixed (continuous and discrete) parent sets via a product-kernel construction, with a Kronecker path for small discrete parent sets and a Hadamard-product path otherwise. The Tetrad Random Fourier Feature (TRFF) score is a complementary BIC-style alternative using penalized Student-t regression with random Fourier features. TRFF offers robustness to heavy-tailed noise and faster runtime than FFML. Empirically, TRFF and FFML exhibit a complementary precision-recall profile: TRFF achieves higher precision while FFML achieves better recall and lower SHD overall. The Fourier Feature Conditional Independence (FFCI) test is a fast nonparametric CI test for mixed data, using ridge residualization in feature space and a Frobenius-norm cross-covariance statistic approximated as a weighted sum of chi-squared variables. Empirically, BOSS+FFML achieves the lowest SHD on nonlinear data, while BOSS+TRFF offers the highest precision. When run through PC-Max, FFCI and RCIT exhibit complementary precision-recall profiles: RCIT is more precise while FFCI achieves better recall and substantially lower SHD, at approximately twice the runtime.

Kernel quadrature can exploit RKHS spectral structure and outperform Monte Carlo on smooth integrands, but optimized quadrature weights are generally signed and may be numerically unstable. We study whether spectral acceleration remains possible when the weights are constrained to be positive, i.e., simplex weights. In the exact-target fixed-pool setting, an evaluated i.i.d. candidate pool of size $N$ is already available and the task is to reweight it so as to approximate the kernel mean embedding. We show that this positive reweighting problem is governed not by the equal-weight empirical average, but by the random convex hull generated by the pool. Our main geometric result shows that the mean of a bounded $d$-dimensional random vector can be approximated by a convex combination of $N$ i.i.d. samples at accuracy $O(d/N)$ with high probability, sharper than equal-weight averaging in the fixed-dimensional regime. We transfer this $d$-dimensional convex-hull approximation to full RKHS worst-case error through an augmented Mercer-truncation argument. The resulting positive-weight KQ bounds consist of a spectral tail term and a finite-sample convex-hull term, yielding Monte-Carlo-beating rates in favorable spectral regimes, including near-$O(1/N)$ rates up to logarithmic factors under exponential spectral decay. We also provide a constructive Frank–Wolfe algorithm that operates directly on the pool atoms, maintains simplex weights, and admits an explicit optimization-error bound.

Unlike MLPs, Kolmogorov-Arnold Networks (KANs) expose explicit learnable edge functions on every connection, enabling mechanistic explanation in time-series forecasting. This paper introduces Temporal Functional Circuits, a framework that transforms KAN edge functions from latent visualizations into faithful, temporally grounded explanations. Built on a gated residual KAN that decomposes forecasts into a linear base and a sparsely activated KAN correction, the framework (i) maps each edge to input lags via output-aware attribution, (ii) ranks edges by learned activation range, and (iii) validates faithfulness through edge-level interventions including zeroing and spline removal. Removing the learned B-spline component while retaining the base SiLU term degrades forecasts, providing evidence that the spline shape itself carries predictive value beyond the base activation. On four synthetic regimes of increasing complexity, the learned gate opens progressively wider as signal complexity grows. On regime-switching signals, gated KAN achieves 59% lower MSE than linear-only models. Across eight benchmarks, the gated architecture is competitive with linear, attention, and MLP alternatives, while providing interpretable edge functions that MLP-based corrections cannot offer.

Differential item functioning (DIF) arises alongside latent population heterogeneity in many applications, and both must be accounted for when assessing measurement invariance. In many practical settings, however, the comparison groups are unobserved and anchor items are unknown. A further challenge is that item response theory models traditionally assume symmetric link functions, yet empirical response processes may exhibit substantial asymmetry. This paper proposes a general framework for jointly analysing impact and DIF under asymmetric item response models. Unobserved group differences are represented by latent classes within a mixture item response model, while item-specific shifts capture DIF effects. Assuming the number of DIF items is relatively small, an $\ell_1$-regularised estimator is used to simultaneously identify the latent classes and select DIF items without requiring observed group labels or pre-specified anchor items. A simulation study evaluates recovery of impact, item parameters, and DIF effects across a range of configurations. The method is illustrated using two empirical applications from educational testing. In one dataset, the selected model reveals both impact and item-level DIF, whereas in the other, the results indicate substantial impact but little evidence of item-level DIF.

Training loss and throughput can hide distinct internal representation in language-model training. To examine these hidden mechanics, we use spectral measurements as practical and operational diagnostics. Using a controlled family of decoder-only models adapted from the modded NanoGPT codebase, we introduce an empirical protocol based on activation covariance and per-sample gradient SVD spectra. This dual-view reveals three empirical findings and one mechanistic explanation. First, batch size acts as a latent determinant of representation geometry: runs that reach equal loss settle into systematically distinct activation spectra. Second, the activation covariance tail measured early in training reliably forecasts downstream token efficiency. Third, movement of the activation spectrum head (leading modes), together with gradient spectra, characterizes underlying learning-dynamics changes, separating learning-side architectural improvements from primarily execution-side gains. These predictive and diagnostic signals persist across the 12-, 36-, and 48-layer model tiers. Finally, a mechanistic model proves the main observations and explains how activation covariance spectra correlate with task-aligned feature learning.

Standard cardiovascular risk calculators, including the Framingham Risk Score and the ACC/AHA Pooled Cohort Equations, estimate the conditional probability P(CHD | SysBP = s) rather than the interventional quantity P(CHD | do(SysBP = s)). When confounding is present, this distinction has direct clinical consequences: observational estimates may systematically overstate the absolute benefit of antihypertensive treatment. We applied Pearl's do-calculus to the Framingham Heart Study Offspring Cohort (n = 4,240; primary analysis on 3,776 complete cases; 574 ten-year coronary heart disease events). A structurally corrected directed acyclic graph (DAG) was specified and evaluated using conditional independence testing. The average causal effect (ACE) of a 20 mmHg systolic blood pressure reduction was estimated by g-computation with bootstrap confidence intervals, corroborated by propensity score matching and inverse probability weighting. G-computation yielded an ACE of 3.40 percent absolute risk reduction (95 percent CI: 2.64 to 4.14), compared with a naive observational estimate of 4.14 percent, corresponding to an approximate 21.8 percent relative overestimation. Conditional average treatment effects were estimated using R-Learner and T-Learner metalearners. These findings suggest that observational cardiovascular risk tools may overestimate the absolute benefit of blood pressure reduction, with implications for clinical risk stratification and prescribing thresholds.

These notes provide a pedagogical introduction to the role of transversality theory in the analysis of statistical degeneracies within the framework of distributional statistical models. The classical question of when a statistical model is well-behaved - in the sense of being identifiable, having non-singular Fisher information, and admitting robust estimation - is reformulated as a question about the geometry of a kernel-induced feature map. Statistical pathologies correspond to geometric degeneracies of this map, and transversality theory provides a precise language for understanding when and why such degeneracies are non-generic. The exposition is organised in three parts. Part I surveys the statistical phenomena that motivate the geometric treatment: representation failure, non-identifiability, moment indeterminacy, singular information, nuisance parameters, and the Behrens-Fisher problem. Part II develops the necessary geometric toolkit - smooth maps, Sard's theorem, transversality, jets, stratifications, and the parametric transversality theorem - at a level accessible to students with a background in analysis and linear algebra but no prior exposure to differential topology. Part~III returns to the statistical problems of Part~I and shows how each one admits a unified geometric interpretation as a transversality condition on the feature map. These notes are a pedagogical companion to the research paper Labouriau (2026) "Transversality and Geometric Regularisation in Distributional Statistical Models" (arXiv:2605.04536 [math.ST]), expanding its arguments with motivating examples, geometric intuition, and exercises aimed at advanced Master's and PhD students with a background in mathematical statistics and measure theory. They are designed to support seminars or reading groups.

We study the problem of learning generative models for discrete sequences in a continuous embedding space. Whereas prior approaches typically operate in Euclidean space or on the probability simplex, we instead work on the sphere $\mathbb S^{d-1}$. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score. The conditional velocity is in general intractable. Exploiting the radial symmetry of the vMF density we reduce the continuity equation on $\mathbb S^{d-1}$ to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on $(\mathbb S^{d-1})^L$ both decompose into posterior-weighted tangent sums that differ only by per-token scalar weights. This gives access to both ODE and predictor-corrector (PC) sampling. The posterior is the only learned object, trained by a cross-entropy loss. Experiments compare the vMF path against geodesic and Euclidean alternatives. The vMF path especially in combination with PC sampling significantly improves results on Sudoku, language modeling, and mathematical reasoning.

We study contextual dynamic pricing with linear valuations and bounded-support agnostic noise, whose induced demand curve may be non-Lipschitz with arbitrary jumps and atoms. Such discontinuities break the cross-context interpolation arguments used by smooth-demand pricing algorithms, while the best previous method achieved only $\tilde O(T^{3/4})$ regret. We propose Conservative-Markdown Redirect-UCB Pricing, a polynomial-time algorithm that combines randomized parameter estimation, conservative residual-grid probing, and confidence-based one-step redirection. Our algorithm achieves $\tilde O(T^{2/3})$ optimal regret, matching the known lower bounds of Kleinberg and Leighton (2003) up to logarithmic factors and improving over the previous upper bound of Xu and Wang (2022). Under stochastic well-conditioned contexts, this closes the long-existing open regret gap in linear-valuation contextual pricing under agnostic non-Lipschitz noise distribution.

Stochastic differential equations (SDEs) provide a flexible framework for modeling temporal dynamics in partially observed systems. A central task is to calibrate such models from data, which requires inferring latent trajectories and parameters from sparse, noisy observations. Classical smoothing methods for this problem are often limited by path degeneracy and poor scalability. In this work, we developed a novel method based on characterization of the posterior SDE in terms of conditional backward-in-time score defined as the gradient of a function solving a Kolmogorov backward equation with multiplicative updates at observation times. We learn this conditional score using neural networks trained to satisfy both the governing PDE and the observation-induced jump conditions, thereby integrating continuous-time dynamics with discrete Bayesian updates. The resulting score induces a posterior SDE with the same diffusion coefficient but a modified drift, enabling efficient posterior trajectory sampling. We further derive a likelihood-based objective for learning the SDE parameters, yielding an evidence lower bound (ELBO) for joint state smoothing and parameter estimation. This leads to a variational EM-style procedure, where the neural conditional score is optimized to approximate the smoothing distribution, followed by a maximization step over the SDE parameters using samples from the induced posterior. Experiments on nonlinear systems demonstrate accurate and stable inference with a very few observations demonstrating significant improved scalability compared to classical MCMC methods.

Express transportation network design is uncertain because origin–destination demand, travel time, operating cost, hub congestion, and realized sorting productivity vary over time. Existing multi-topology express network models usually optimize cost and maximum arrival time under fixed input data, which may produce designs that are efficient nominally but fragile under demand surges, route disruptions, and hub productivity losses. This paper develops a Bayesian posterior-predictive framework for multi-topology express transportation network design. The model learns demand, travel-time, cost, and hub-reliability uncertainty from historical or benchmark-calibrated data and propagates them through posterior predictive scenarios. For fully connected, hub-and-spoke, restricted-allocation, and direct-link hybrid topologies, candidate designs are evaluated using posterior expected cost, conditional value-at-risk of maximum arrival time, service reliability, hub hold-time reliability, and emission-aware penalties. A Bayesian multi-structure design methodology is proposed using posterior simulation, sample-average approximation, topology-wise optimization, and Bayes-risk selection. Theoretical results establish existence of a Bayes-optimal design, convergence of posterior scenario risks, and stability of topology selection. Simulation and CAB benchmark experiments show that the Bayesian design can trade modest additional cost for substantial reductions in tail delivery risk and improved hub reliability.

Positive-unlabeled (PU) learning addresses binary classification when only a set of labeled positives is available alongside a pool of unlabeled samples drawn from a mixture of positives and negatives. Existing PU methods typically require dataset-specific training or iterative optimization, which limits their applicability when many tasks must be solved quickly or with little tuning. We introduce PUICL, a pretrained transformer that solves PU classification entirely through in-context learning. PUICL is pretrained on synthetic PU datasets generated from randomly instantiated structural causal models, exposing it to a wide range of feature-label relationships and class-prior configurations. At inference time, PUICL receives the labeled positives and the unlabeled samples as a single input and returns class probabilities for the unlabeled rows in one forward pass, with no gradient updates or per-task fitting. On 20 semi-synthetic PU benchmarks derived from the UCI Machine Learning Repository, OpenML, and scikit-learn, PUICL outperforms four standard PU learning baselines in average AUC and accuracy, and is competitive on F1-score. These results show that the in-context learning paradigm extends naturally beyond fully supervised tabular prediction to the semi-supervised PU setting.

Despite the growing availability of large datasets, causal structure learning remains computationally prohibitive at scale. We revisit sparsest-permutation learning for linear structural equation models and show that exact Cholesky factorization is unnecessary for structure recovery. This observation motivates a support-level relaxation that searches for sparse triangular factors over a precision-support screening graph. The relaxed formulation can be efficiently evaluated via masked zero-fill incomplete Cholesky factorization, enabling scalable comparison of candidate orderings. At the population level, we establish soundness for Markov equivalence class (MEC) recovery under no-cancellation and sparsest Markov representation assumptions, as well as robustness to ordering misspecification. Motivated by these guarantees, we introduce SCOPE, a sparse-Cholesky pipeline that provides a scalable implementation of the relaxed formulation. Experiments on synthetic and real datasets demonstrate that SCOPE matches the MEC recovery accuracy of substantially slower baselines, while achieving significantly reduced runtime and scaling to 10k variables.

Machine-learning systems used in survey-based social measurement require uncertainty estimates that are reliable across population subgroups, not merely valid in aggregate. We study ordinal conformal prediction for five-level AI-attitude forecasting on the Pew American Trends Panel (Wave 152; n=4,591; 12 race x education subgroups), comparing standard split conformal, Mondrian (group-specific) conformal, and a regularized Mondrian comparator across 100 respondent-disjoint splits with survey-weighted evaluation. Standard conformal achieves nominal marginal coverage for all four base predictors but leaves weighted subgroup gaps of ~13 percentage points. For the strongest predictor (XGBoost), Mondrian worsens the fairness-efficiency trade-off: weighted set size rises by +0.036 (dz =1.66) while the weighted subgroup gap grows by +0.013 (dz =0.30). A regularized comparator that shrinks group thresholds toward the global quantile mitigates this instability (Delta gap = -0.001, Delta size = +0.012) but does not yield a decisive fairness gain. Failure analysis traces the mechanism to calibration-cell fragmentation interacting with group-specific confidence mismatch. The negative result persists across alternate outcome codings and subgroup granularities, demonstrating that nominal marginal validity is insufficient for subgroup reliability and that naive group-specific calibration is not a dependable fairness remedy in complex survey settings.

This paper introduces the Statverse, a Metaverse framework designed to revolutionize statistical education in the digital age. Our key goal is to report our progress and encourage others to integrate similar strategies into their programs. The proposed framework seamlessly integrates the physical and digital realms to provide an immersive environment for the nuanced representation of complex statistical concepts. Finally, we discuss the potential impact of Statverse on advancing Statistical Education, offering a transformative approach to teaching and learning in the digital age. Statverse is the outcome of an academic partnership between Universidad Técnica Federico Santa María (UTFSM) and the University of Edinburgh (UoE).

Sparse regression based on global-local shrinkage priors are increasingly used for Bayesian modeling of modern high-dimensional data, but scaling up the Gibbs sampler for posterior inference remains a challenge. While much effort has gone into speeding up the high-dimensional coefficient update step, insufficient attention has been given to the potential poor mixing of the global scale parameter $\tau$ and of the overall sampler. One proposed remedy has been to marginalize out the coefficients when updating $\tau$. Here we show that, while this collapsed update was previously thought to require a Metropolis step, we can in fact sample directly and efficiently from the collapsed density. This is made possible by careful linear algebraic manipulations and a strategic per-Gibbs-scan spectral decomposition, allowing subsequent evaluations of the collapsed density across hundreds of values of $\tau$ at negligible cost. We combine this computational trick with adaptive numerical integration and inverse transform sampling to construct a direct sampler. This eliminates the need to tune Metropolis proposals and yields faster convergence and improved mixing. We demonstrate our method on two big data applications, fitting logistic regression under the horseshoe prior to datasets with design matrices of size 120,000 x 1,379 and 1,980 x 17,848.

Suppose that a sequence of data points follows a distribution of a certain parametric form, but that one or more of the underlying parameters may change over time. This paper addresses various natural questions in such a framework. We construct canonical monitoring processes which under the hypothesis of no change converge in distribution to independent Brownian bridges, and use these to construct natural goodness-of-fit statistics. Weighted versions of these are also studied, and optimal weight functions are derived to give maximum local power against alternatives of interest. We also discuss how our results can be used to pinpoint where and what type of changes have occurred, in the event that initial screening tests indicate that such exist. Our unified large-sample methodology is quite general and applies to all regular parametric models, including regression, Markov chains, and time series situations.

Background: Fentanyl overdose deaths are still increasing across the U.S. We do not fully understand which county-level social and structural conditions lead to higher overdose death rates. Social determinants of health, including disability, treatment access, and behavioral health issues, may help identify vulnerable counties before deaths become severe. No earlier study has used explainable machine learning with SHAP attribution on 2022 CDC WONDER data to study treatment access gaps and silent risk counties. Methods: We combined data from four government sources for 975 U.S. counties, including CDC WONDER (2022) overdose mortality data, CDC Social Vulnerability Index (SVI), CDC PLACES health behavior data, and Area Health Resources Files. An XGBoost model was used to predict overdose mortality risk using Standardized Mortality Ratio (SMR). Five-fold cross-validation was used to test model accuracy, and SHAP values were used to show which factors increase or decrease risk. Results: XGBoost outperformed all tested models (Spearman rho=0.67, R2=0.457, MAE=0.409, high-risk recall=71.1%). Top predictors were disability rate, hypertension, smoking, and lack of vehicle access. Treatment desert counties had 52.6% higher overdose mortality (SMR 1.786 vs 1.170; p<0.0001). K-means identified 143 silent risk counties. Overdose deaths were spatially clustered (Moran's I=0.505, p=0.001) with 75 hotspots and 136 coldspots. Suppressed counties were 58.2% of WONDER counties, mostly rural (72%) and treatment deserts (65%). Conclusions: County-level SDOH factors predict overdose deaths, especially disability, treatment access, and behavioral health burden. MOUD expansion should prioritize treatment desert counties, and silent risk counties need early intervention before mortality worsens.

Information theory plays a central role in establishing fundamental limits on what any learning or estimation algorithm can – and cannot – achieve, regardless of computational power. In this chapter, we provide an introduction to these connections. End-of-chapter exercises makes the material suitable for both classroom use and self-study. We begin by introducing concentration inequalities along with the notions of covering and packing in metric spaces, and the associated concept of metric entropy. These tools are essential for our analysis. We then introduce the learning-theoretic framework and derive upper bounds on generalization error in terms of metric entropy, Rademacher complexity, and the VC dimension, as well as mutual information and relative entropy. Finally we discuss the minimax estimation framework and establish lower bounds on minimax risk using Fano's inequality, yielding bounds in terms of relative entropy and covering and packing numbers. This manuscript contains preprint of a chapter under consideration for inclusion in the forthcoming third edition of Cover and Thomas's Elements of Information Theory, posted with permission from Wiley. It would follow the chapter posted at arXiv:2605.02989 . The table of contents of the new edition can be found at: https://docs.google.com/document/d/1L-m4oQEJw1PJhoxBeMwrrBD8S_HmvzMEkPbYvS24980/edit?usp=sharing . For feedback, please contact abbas@ee.stanford.edu.

We introduce a novel framework for constructing scalable and flexible covariance kernels for Gaussian processes (GPs) by directly learning the covariance structure under a regression-type parameterization induced by Vecchia approximations, using deep neural architectures. Specifically, we model kriging coefficients and conditional standard deviations, deterministic quantities that uniquely characterize the covariance, providing stable and informative learning targets. Exploiting the permutation-equivariant structure of conditioning sets in the Vecchia factorization, we derive a universal representation for permutation-preserving functions and design neural architectures that respect this symmetry, leading to improved training stability and data efficiency. The proposed approach enables expressive, non-stationary kernel learning while maintaining computational scalability, thereby bridging classical GP methodology with modern deep learning.

Counterfactual utilities evaluate decisions not only by the realized outcome under a given decision, but also by the counterfactual outcomes that would arise under alternative decisions. By generalizing the standard utility framework, they allow decision-makers to encode asymmetric criteria, such as avoiding harm and anticipating regret. Recent work, however, has raised fundamental concerns about the coherence and transitivity of counterfactual utilities. We address these concerns by extending the von Neumann-Morgenstern (vNM) framework to preferences defined on the extended space of all potential outcomes rather than realized outcomes alone. We show that expected counterfactual utility satisfies the vNM axioms on this extended domain, thereby admitting a coherent preference representation. We further examine how counterfactual preferences map onto the realized outcome space through menu-dependent and context-dependent projections. This axiomatic framework reconciles apparent inconsistencies highlighted by the Russian roulette example in the statistics literature and resolves the well-known Allais paradox from behavioral economics. We also derive an additional axiom required to reduce counterfactual utilities to standard utilities on the same potential outcome space, and establish an axiomatic foundation for additive counterfactual utilities, which satisfy a necessary and sufficient condition for point identification. Our results show that the use of asymmetric counterfactual utility functions is coherent regardless of whether individual potential outcomes are deterministic or stochastic, and that a recent proposal involving stochastic potential outcomes leads to ambiguities and inconsistencies.

Commercial Microwave Links (CMLs) offer dense spatial coverage for rainfall sensing but produce path-integrated measurements that make accurate ground-level reconstruction challenging. Existing methods typically oversimplify CMLs as point sensors and neglect line integration relating rainfall to signal attenuation, resulting in degraded performance under heterogeneous precipitation. In this work, we view rain field reconstruction as a Bayesian inverse problem with Diffusion Models (DMs) as high-fidelity spatial priors. We show that diffusion models better preserve key rainfall statistics compared to censored Gaussian processes. Framing rainfall estimation as a Bayesian inverse problem with a DM prior enables training-free posterior sampling using a broad family of methods, including Plug-and-Play, Sequential Monte Carlo, and Replica Exchange methods. Experiments on synthetic and real-world datasets demonstrate consistent improvements over established CML-based reconstruction baselines.

Active feature acquisition (AFA) considers prediction problems in which features are costly to obtain and the learner adaptively decides which feature values to acquire for each instance and when to stop and predict. In this paper, we introduce a continuous relaxation of the acquisition process that enables non-myopic pathwise policy gradients (NM-PPG) through the full acquisition trajectory, avoiding the high variance of standard score-function policy gradients while allowing end-to-end optimization of the acquisition policy. To better align training with deployment, we develop a straight-through rollout that follows discrete feature acquisitions in the forward pass while backpropagating through the corresponding soft relaxation. We derive an AFA-specific average-case upper bound on the variance of this gradient estimator, which characterizes instability and motivates staged temperature sharpening. Experiments on both synthetic and real-world datasets demonstrate that NM-PPG yields superior performance relative to state-of-the-art AFA baselines.

We formally introduce a class of models inspired by renormalization group (RG) theory, built on additive hierarchical expansions analogous to those appearing in functional ANOVA and mixed-effects models. Like ReLU convolutional neural networks, they are almost everywhere locally linear; unlike ReLU networks, their partition structure is explicit, interpretable, and easy to modify or constrain. In these models, one defines a multidimensional lattice partition of the input space and uses it to scaffold variations in regression parameters. Each dimension of the lattice corresponds to an attribute by which the statistics of the problem may vary. The parameters are themselves expressed in the form of an expansion, where each term captures variations relative to a lower (coarser) interaction scale. These models admit multiple equivalent interpretations: as piecewise GLMs, as hierarchical mixed-effects regressions, or as regression trees with structured parameter sharing. Since RG motivates the design of these models, we use techniques from statistical physics – specifically replica analysis – to study their generalization properties. Specifically, we analyze the behavior of the Watanabe-Akaike Information Criterion (WAIC) as a proxy for generalization loss. This analysis yields two practical results: (i) guidance on the lattice design as a function of dataset size and predictor dimensionality; and (ii) a principled scaling law for the regularization prior when adding higher-order terms to the expansion so that one can increase model complexity without an expected increase in generalization loss. We evaluate the methodology on public datasets and find performance competitive against both blackbox methods and other intrinsically interpretable approaches.

The main post-hoc XAI methods for deep networks – GradCAM, SHAP, LIME, Integrated Gradients – originate from heterogeneous theoretical foundations and are not naturally comparable within a single representation. A recent benchmark also finds their coalition-based members (GradCAM, KernelSHAP, LIME) and gradient-based members (Integrated Gradients and variants) empirically complementary, each outperforming the other on different faithfulness metrics, with method selection as the only proposed remedy (Gevaert et al., 2022). This work presents GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), which fuses these two mechanisms – a Shapley coalition weight and locality kernel, and a continuous Integrated-Gradients-style conditioned path – into a single estimator, and equips it with two certified guarantees neither mechanism supplies alone: an exact, closed-form completeness deficit (an order-d interaction is attributed at a factor 1/d of its true value under Shapley weights and a multilinear F) and a finite-sample bound, O(1/sqrt(m)) + O(1/k^2), for the actual self-normalized ratio the algorithm returns. This fusion is underpinned by a representation-theoretic result: every additive, linear, continuous attribution functional admits a unique canonical representation via the Riesz Representation Theorem, proved componentwise (feature by feature) rather than as one form shared across features or methods. This class includes SHAP, IG and LIME, but not nonlinear functionals such as standard GradCAM or attention maps. Seven theorems further establish an exact correspondence with Shapley Interaction Values, affine-regime correspondences with the Hoeffding/Sobol decomposition, and a minimum-variance multi-scale extension. A preliminary experimental illustration on breast histology imaging is included; extended validation is in a companion paper (Fanale, 2026).

High-dimensional functional data are becoming increasingly common in fields such as environmental monitoring and neuroimaging. This paper studies high-dimensional functional linear regression models that relate a scalar response to ultra-high-dimensional functional predictors, where each predictor is treated as a random element in an infinite-dimensional functional space. To address the dual challenges of high-dimensionality and model interpretability, we propose MoFI-FLR, a novel two-step estimation framework rooted in reproducing kernel Hilbert space (RKHS) theory. The first step employs a functional elastic-net penalty to screen out irrelevant covariates, while the second step decomposes each selected predictor's functional coefficient into an interpretable finite-dimensional simple component and an infinite-dimensional complementary complement. By penalizing only the complementary component, our method automatically distinguishes simple effects, which consist only of the simple component, from complex effects, which also include complementary deviations. Under mild regularity conditions, we establish non-asymptotic theoretical guarantees, demonstrating that MoFI-FLR consistently recovers the active covariates and accurately identifies their true functional forms. We develop a computationally efficient algorithm to implement the proposed method and evaluate its performance through comprehensive simulation studies and an application to Psychomotor Vigilance Task EEG data.