Papers

We introduce an efficient Bayesian deep ensemble method for predictive regression designed to enhance interpretability while maintaining competitive predictive performance and computational efficiency. Our method combines the statistical rigor of Bayesian inference with the scalability of deep ensembles, providing calibrated uncertainty estimates that enable its use not only for standalone prediction but also as a component within broader learning systems. To achieve these goals, our work relies on three key design components: (i) low-dimensional ensemble representation: predictions are expressed as a combination of a small number of trained neural predictors, enabling scalable inference whose cost depends on ensemble size rather than dataset size; (ii) closed-form Bayesian aggregation: ensemble predictions are combined using Bayesian linear regression, yielding interpretable posterior weights and calibrated uncertainty without approximate inference; and (iii) Independent ensemble training: multiple neural networks are trained separately, producing diverse predictive representations that improve robustness and uncertainty calibration. Empirical results on standard regression benchmarks demonstrate that the proposed approach achieves competitive predictive performance while maintaining reliable uncertainty estimates across settings.

Spacing, the difference between consecutive order statistics, has two features that reflect the modality of the data. Consistent, stable values occur around modes while local increases mark the transitions between them. These features not only signal multi-modality, they also locate modes and anti-modes. Dimodal is an R package for detecting and evaluating these situations. It includes parametric feature models and bootstrap tests for spacing smoothed by low-pass filtering, non-parametric runs and permutation tests for the interval spacing, and a fusion of changepoints in the raw spacing. We introduce the analysis, describe the package, its implementation and performance, and apply it to identifying Kirkwood gaps in the asteroid belt. We also present ports of the software, with DimodalCPy a command-line program written in C with a Python interface.

In this article, we address the problem of estimating a forcing parameter in a stochastic differential equation inspired by a model that describes instantaneous turbulent kinetic energy. The stochastic differential equation we analyze is of the nonlinear McKean-Vlasov type, where the drift term depends on a power of the expected value of the solution, which also introduces nonlinearity in an algebraic sense. We propose an estimation algorithm based on the Expectation-Maximization framework and show the consistency of our method. We illustrate our findings through numerical experiments.

Weighted empirical measures on compact manifolds appear in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Ordinary effective sample size and related weight summaries ignore the geometry of the support. We introduce heat-kernel entropy profiles to measure nonuniformity after intrinsic diffusion at a range of scales. For order-two Rényi entropy, pairwise heat-kernel overlaps give an exact profile and a geometric effective sample size. This effective sample size discounts nearby or duplicate particles. It approaches ordinary effective sample size as overlaps between distinct particles vanish. On compact boundaryless manifolds, we establish profile monotonicity, gESS scale limits, deterministic-weight consistency, and a bounded-ratio result for self-normalized importance sampling. On spheres, the unlogged profile decomposes into spherical-harmonic energies. The first terms are squared mean-resultant and traceless-second-moment energies, which give vMF- and Bingham-type scalar summaries. Experiments identify antipodal, girdle, multimodal, and duplicate-particle structures that weight-only and first-moment summaries miss.

Finance, sensing, and demand streams violate the exchangeability that IID conformal prediction and the IID bootstrap assume, and existing libraries implement either a general resampling engine or conformal calibration without the other. tsbootstrap provides block, residual, sieve, and wild resampling, classical bootstrap confidence intervals, and adaptive conformal calibrators (EnbPI, ACI, NexCP, AgACI) through a single typed API in which a specification object selects each method. In a controlled coverage study the IID bootstrap undercovers sharply under dependence; dependence-aware methods reduce the coverage deficit, the sieve nearest to nominal under short-memory linear dependence. On the shared fixed-statistic path a compiled backend runs several times faster than arch, and a streaming reduce avoids materializing the $O(Bn)$ replicate tensor, limiting peak extra memory to $O(B)$ for the statistic array. The software is MIT licensed (v0.6.1).

Knowledge workers switch between applications thousands of times per day, spending nearly a tenth of the work year transitioning between digital applications in a process called digital fragmentation. Whether this fragmentation reflects who an employee is, where they work, or what kind of day they are having, has remained an open question. We analyzed 103 million application events recorded second-by-second from 1,017 employees across eight organizations that largely employ knowledge workers (e.g., law, financial services). Day-to-day variation in fragmentation within individual employees accounted for 44.6% of the variation in digital fragmentation, slightly exceeding stable individual differences between employees (35.8%), and far exceeding variation between organizations (19.6%). Fragmentation rose over the work week and reset after weekends and holidays. Higher-than-typical use of communication applications coincided with more fragmented work. Generative AI use also occurred on more fragmented days, but the period following AI use was marked by narrower, longer, and more predictable application use. These findings identify the workday as a key level for understanding and intervening on digital fragmentation and suggest that AI may help structure fragmented work rather than merely intensify it.

Rough path signatures are a universal feature map for continuous paths and, via the expected signature, characterise path distributions. These guarantees do not directly extend to cadlag paths of Temporal Point Processes (TPPs), limiting the use of signature methods for event sequences. Furthermore, neural TPP models, including recent generative approaches, optimise per-event objectives with no global sequence-level loss, while evaluation of variable-length event sequences lacks distributional discrepancy measures. This paper proposes a common pathwise framework for addressing these limitations. We introduce the interarrival embedding, a stable, injective lift from jump paths to continuous paths of bounded variation, extending signature methods to discrete event sequences. Our theoretical contributions give rise to sigTPP, the first signature-based generative model for TPPs, trained using a path-level loss on complete trajectories. We further analyse the space of counting paths and derive three distributional discrepancies, providing mathematically justified tools for evaluating generative TPP models. Across synthetic and real-world datasets, sigTPP achieves the best average rank based on eight complementary metrics, outperforms or is within a standard error of the strongest baseline in 64% of the dataset-metric pairs, and according to a relative score, improves against every baseline by at least 19% on average.

Determinantal point processes have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficient minibatches, coresets, and other compact representations of large-scale datasets. In particular, sampling mechanisms based on DPPs are believed to demonstrate better approximation properties compared to classical i.i.d. samplers, even at the scale of the exponent. One of the key strengths of DPP based samplers is that they can be deployed over very general spaces, in contrast to more classical sampling methods beyond i.i.d. which tend to work in very well-structured settings, principally Euclidean spaces. In this work, we establish explicit rate guarantees for determinantal sampling in spaces that extend far beyond known Euclidean setups, focusing on spectral kernels obtained from eigenspaces of naturally associated Laplacian and other Markov diffusion operators. This includes, in particular, Riemannian manifolds and weighted networks. In determinantal sampling from compact Riemannian manifolds, we establish sampling rates that automatically pick up the intrinsic dimensionality $d_{\text{int}}$ of the underlying manifold. In the setting of networks, we investigate DPP-based samplers on the celebrated k-nearest neighbour graphs, as well as weighted random geometric graphs, and demonstrate a similar improved dependence on the intrinsic dimensionality of the data. Overall, our approach achieves guarantees of $\big(\text{sample size}\big)^{-\frac{1}{2}-\frac{1}{2d_{\text{int}}}}$ that match known rates on Euclidean spaces of comparable dimension. In terms of techniques, we connect to the celebrated Weyl's Law for manifold spectra, and leverage tools from the theory of Markov diffusions and Dirichlet forms as well as certain ingredients from the theory of pseudodifferential operators, which could be of independent interest in this area.

In this work, we propose a unified approach for diagnosing misclassification and assessing the robustness of black-box classifiers. Central to our method is an optimization framework that modifies an instance so that the classifier predicts a specified target label, while ensuring that the modification remains easily explainable. The objective function contains two components: an explainability-aware $L_0$ (XA-$L_0$) penalty that promotes sparse and interpretable modifications, and a classifier loss objective that steers the perturbed instance toward the desired output. This integrated optimization formulation is used both to identify the underlying causes of misclassification and to evaluate robustness by determining how an instance can change within a tolerance region before being reassigned to another class. To quantify robustness, we introduce the Tolerance Region Confusion Matrix (TOR-Confusion Matrix), which measures a classifier's susceptibility by modeling the class-to-class transition probabilities induced by tolerance-bounded perturbations. We validate the proposed method on both image and tabular datasets, demonstrating its ability to jointly deliver interpretability and robustness assessment.

Conformal prediction is being adopted in drug discovery to put an honest number on model reliability: pick an error rate alpha, and the method returns prediction sets containing the true label with probability at least 1 - alpha. We show this guarantee can be dangerous on imbalanced datasets. Across four datasets, standard (marginal) conformal prediction hits its global 90% coverage target while leaving the minority class badly exposed: realized minority coverage falls to 64.8% on blood-brain-barrier penetration and to 4.2% on clinical-trial toxicity, where the rare class is nearly abandoned. The failure is not tied to one model: a random forest, a graph network, and a frozen chemical language model all reproduce it (p < 0.001 in every case), with severity tracking baseline calibration on rare labels rather than architecture. A conservation identity explains the effect: the minority's shortfall equals the majority's surplus amplified by the imbalance ratio, predicting the measured gap to within one point and ordering severity across datasets. The failure survives realistic scaffold splits and a second conformal score, while aggregate accuracy and overall coverage stay reassuringly high, which is exactly why it is easy to miss. Class-conditional (Mondrian) conformal prediction closes the gap on every dataset, restoring minority coverage to target for a modest increase in prediction-set size. We localize the failures to generic molecular scaffolds - plain benzene and pyridine cores occurring in both classes - propose a one-number diagnostic, and show with a cost model that abstaining on affected compounds flips a screening campaign from net-negative to net-positive utility. Our contribution is demonstrating on real chemistry how severe and invisible this known conformal-theory gap becomes under imbalance, and laying out a practical protocol restoring per-class reliability.

Interventions delivered in space generate effects that spill over between experimental units. We develop a framework for spatial experiments in which causal estimands, inclduing direct, indirect, total, and dose–response effects, are linear functionals of a spillover kernel. Under cluster randomisation with a fixed exposure set size, the data identify the shape of the kernel but not its level: the level is aliased with the intercept and direct effect, at any sample size and under any outcome model. Every estimator therefore rests on an anchoring assumption that fixes the level. We derive an exact decomposition of the bias of any anchored estimator into a level term, a shape error from kernel misspecification, and a leakage term absorbed by the realised geometry, each computable from the design before data collection. A single scalar, the level leverage, gives each estimand's exposure to the unidentified level, the exact level bias, and the variance cost of estimating the level instead. The conventional cluster-trial analysis is the special case of an implicit anchor, with contamination bias in closed form. Existing approaches, including identification through Bernoulli randomisation, elicited bounds on interference decay, and assumed compact support, are, within the linear exposure mapping, anchoring choices in this framework. We give a taxonomy of design augmentations that purchase the level and a criterion for when to augment and when to anchor.

We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths. Many existing graph-based embeddings rely either on locally normalised random walks or on shortest-path distances. The former can concentrate diffusion in densely sampled regions, while the latter are sensitive to spurious shortcut edges in the graph. EntroPath instead builds its dissimilarities from the maximum entropy random walk (MERW), which aggregates the full ensemble of k-step paths between points rather than relying on any single trajectory. We show that the resulting free-energy dissimilarity converges to squared geodesic distance in the short-time limit, via Varadhan's heat-kernel formula. The diffusion depth k interpolates smoothly between local neighbourhood structure and global manifold geometry, and the symmetrised kernel admits an exact Gram factorisation connecting EntroPath to kernel methods. We further provide scalable extensions via landmark projection and diffusion-potential pseudotime. Across synthetic manifolds and single-cell benchmarks, EntroPath consistently matches or outperforms diffusion- and shortest-path-based methods, while remaining competitive with neighbourhood-preserving embeddings (UMAP, t-SNE) on local-structure metrics. Its gains are most pronounced on manifolds with non-uniform sampling density and well-separated branching trajectories, where path-ensemble diffusion more faithfully preserves the underlying geodesic geometry.

Live sports commentary is grounded generation under a deadline: statements concern real, named athletes, the grounding state changes every few seconds, and no reference text exists at generation time. We present Pitwall, a production system that generates natural-language Formula 1 strategy briefings in English, Spanish, and Portuguese, treating faithfulness as an architectural property rather than an aspiration: every published sentence is decomposed into typed factual claims (positions, gaps, tyres, pace, overtakes, race control) and each claim is verified against the probabilistic race state that prompted it. The same verifier gates the fine-tuning data: of 3,045 model-written targets, only the 81.9% whose every claim is state-supported are retained, the rest falling back to a provably faithful template, so the generator never sees an ungrounded target. Verification is meaningful because of the grounding substrate: a vectorized Monte Carlo engine (N=2,000 per-lap race continuations) calibrated on 126 races (2018-2024) and validated on fully held-out 2025-2026 seasons (winner-in-top-3 90.3% over 155 backtests; held-out Brier 0.0745). A recurring finding spans both halves of the system: virtues trade off and must be gated separately. In simulation, calibration-optimal is not decision-optimal; in generation, fine-tuning on richer targets buys vividness that collapses into hallucination when the grounding state is sparse – a failure a four-base replication traces to base-model instruction adherence, not scale, and that sparse-context auditing removes from the production model. End-to-end operation – live timing to verified trilingual briefings – was confirmed at two consecutive live Grands Prix (Austria and Britain, 2026); at Silverstone a timestamped probability trace, committed to disk before the outcome was known, locked onto the eventual winner ten laps before the flag.

Large language model coding agents increasingly perform open-ended data modeling and analysis. These agents are stochastic and adaptive, and therefore their autonomous model discovery behavior cannot be adequately characterized by a single benchmark run. In this work, we propose an experimental design and analysis framework for systematically evaluating this discovery process, quantifying its variability, and identifying important factors. The proposed framework treats these agents as stochastic model-discovery operators, which map task-specific discovery data and an optimization target to a fitted model. Specifically, we investigate two such operators, Codex and Claude Code, under controlled experimental factors including agent's reasoning effort, task, optimization metric, and composition of training data. For each agent-task-metric combination, regression models and inference are conducted for multiple responses such as output quality, dollar cost, wall-clock time, and process complexity. Furthermore, we develop a utility-aligned canonical decomposition to characterize the dominant direction of the reasoning-effort effect and to assess whether that direction aligns with a performance-cost utility direction. The proposed framework is demonstrated on a testbed of networked word-forming games with insightful findings on reasoning effort with respect to cost and process complexity.

A persistent empirical observation is that trained neural networks outperform their neural tangent kernel (NTK) limit on tasks with compositional structure, yet a quantitative account of $\textbf{when}$ and $\textbf{by how much}$ has been lacking. Working on the unit circle, we give such an account through a dichotomy between two complexity measures of the target: its $\textbf{Fourier complexity}$, which controls NTK kernel regression, and its $\textbf{architectural complexity}$, which controls learning over depth-$L$, width-$w$ ReLU networks with the variation norm of the weights bounded by $R$. We first characterize the minimax rate of the architecture class $\mathcal{C}_{L,w,R}$, pinning it down up to a single factor of $L$: between $\Omega(Lw^2R^2/n)$ and $\tilde{O}(L^2w^2R^2/n)$. We then show the NTK estimator sits $\textbf{exponentially}$ above this floor whenever the two complexities decouple: for the depth-$L$ iterated sawtooth, NTK regression needs $\Omega(4^L)$ samples while the minimax floor is polynomial in $L$. Numerical experiments confirm the theoretical claims: on bandlimited smooth targets, the NTK is competitive or better, while on the hypercube sparse-parity model, a standard two-layer network beats the NTK by four to six orders of magnitude in test error. The gap is thus a function-space property, a mismatch between the kernel's smoothness bias and the target's compositional structure, rather than a generic kernel-versus-network phenomenon.

Rolling covariance estimates feed two objects that are routinely treated as market structure. The first is the dominant eigenspace, monitored through the projector movement $\widehat D_{K,t}=\|\widehat P_{K,t}-\widehat P_{K,t-1}\|_F$; the second comprises scalar spectral functionals such as the absorption ratio and the leading-eigenvalue share. Both fluctuate under estimation noise, and shrinkage changes the law of that noise, so reading their movements as structural change requires calibration. For the eigenspace, we derive a first-order null law for $\widehat D_{K,t}$ between overlapping windows that share most of their data and show that it transfers without change to rotation-equivariant shrinkage estimators. A distribution-free Davis-Kahan band gauges whether the eigenspace is identified, an estimator-aware bootstrap provides the calibrated test, and a companion power analysis gives an approximate design rule for the smallest detectable rotation. For the scalar functionals, we show that first-order immunity to elliptical kurtosis holds for scale-invariant functionals and only for them, so that one estimated scalar calibrates the projector null and the absorption-ratio and leading-share intervals across the elliptical family. In high dimensions, where shrinkage cleaning biases the absorption ratio, we give a trace-preserving spike-debiased estimator that removes the bias. The results are verified by simulation under a known population covariance; an equity-panel appendix shows the procedures as diagnostics when the population is unknown.

This paper investigates feature learning within the framework of the deep Ritz method for solving the stationary Schrödinger equation with Neumann boundary conditions. We first analyze the convergence of Riemannian gradient descent in an agnostic setting, where the hypothesis function is restricted to a single-index model while the PDE solution is arbitrary. We prove that gradient descent reaches an approximate global minimum: after T = O(log(1/\epsilon)) iterations, the loss is within \epsilon of a constant multiple of the optimal loss. We then examine the loss landscape when the source term of the PDE itself follows a single-index model, considering hypothesis functions given by either a single-index model or a two-neuron multi-index model. In the single-index case, we show that the minimum Ritz energy is attained at the feature vector aligned with that of the source term. In the two-neuron case, we study the landscape of regularized Ritz losses and characterize how a second feature emerges, given that the first feature is aligned with the source, as the regularization parameter varies. Finally, numerical experiments are presented to validate the feature emergence theory in the two-neuron setting.

This paper develops the asymptotic theory for high-dimensional panel data regressions in settings with cross-sectionally dependent errors driven by common shocks. We consider a factor-augmented sparse-group LASSO estimator that combines MIDAS aggregation with latent factors. The estimator can take advantage of the mixed-frequency group structure in the time-series dimension. Theory shows that it can outperform the standard LASSO estimator both for prediction and estimation while allowing for cross-sectional dependence.

We develop an approximate risk minimization framework for shrinkage-thresholding estimation in normal mean problems. In the canonical multivariate normal mean model, we introduce a general functional class of estimators that contains classical shrinkage and thresholding behavior, including James-Stein-type and lasso-type rules. We express quadratic risk as a functional over this class, derive optimality conditions for both oracle risk and data-driven approximate risk minimization, and construct a feasible approximate risk criterion from the observed data when the oracle risk is unavailable. The resulting estimator, NOMAD, is obtained by minimizing this approximate risk over the proposed class. For the canonical model, we develop an approximate risk minimization theory that includes optimizer characterization, sieve-based consistency under regularity conditions, and approximate-risk inequalities relative to benchmark procedures in the admissible class. We then extend the framework to multivariate normal mean estimation with correlated observations, develop both MLE-based and conditional MLE-based constructions, and establish consistency results under regularity conditions. We further apply the framework to linear regression and derive an equivalent penalized regression representation in which the shrinkage-thresholding map induces a data-adaptive penalty, recovering ridge-type and lasso-type behavior as special cases or limiting forms. The results provide a unified risk-based framework for shrinkage, thresholding, and regularization across canonical and correlated normal mean estimation and linear regression.

We study the estimation of bilinear forms from noisy, partially observed multilayer data. The signal follows a Tucker2 model, with shared unit and time factors across tensor layers and slice-specific cores. The missingness pattern is structured and motivated by staggered adoption designs, which are common in causal inference and related applications. We first analyze the four-block missingness pattern, the basic building block for general staggered adoption, and propose a spectral algorithm that pools information across layers and targets the functional directly. We prove a non-asymptotic mean squared error bound that exhibits a phase transition in the number of layers, showing when pooling improves estimation, and match it with a local minimax lower bound up to constants when ranks and logarithmic factors are treated as constant-order quantities. We then extend the construction to general staggered adoption designs via an anchored four-block reduction, and derive analogous theoretical guarantees. Finally, we validate our theoretical findings using synthetic and real-world data on Castle Doctrine laws and COVID-19 policies

The use of Gaussian processes for approximating differential equations has expanded rapidly, leading to a growing, diverse, and fragmented body of numerical methods. We present a unified Bayesian perspective that places these techniques within a common probabilistic framework, based on a derivative matching interpretation for incorporating differential equation constraints into likelihood. This unified perspective supports both parameter estimation and solution approximation, and shows how a range of existing methods can be understood within it. This work aims to consolidate current developments and provide a foundation for future research.

We study the infinite-width Gaussian-process limit of random neural networks through the lens of tensor programs, and we provide a quantitative convergence theory in Wasserstein distance. Our main result gives explicit finite-width error bounds, of order inverse square-root of the widths between finite-network executions and their Gaussian-process limits. The framework is architecture-agnostic and covers feed-forward models together with weight-sharing schemes relevant for recurrent and transformer-type architectures.

Information Processing Pathway Maps (IPPMs) offer a scalable framework for formalizing the complex sequence of mathematical transformations applied to sensory stimuli. These maps chart the latency and cortical expression of computational steps, relying on statistical inference to link model outputs with observed neural activity. Traditionally, this mapping has relied on frequentist hypothesis testing. However, determining which of several competing computational models best explains neural data is a problem of model adjudication, arguably better suited to probabilistic inference. Here, we present a direct comparison between the established frequentist approach and a novel Bayesian framework for mapping cortical entrainment. While the Bayesian formulation retains the core strength of IPPMs – generating explicit predictions of time-varying neural signals – it fundamentally alters the selection criterion, shifting from rejecting a null hypothesis to quantifying the relative evidence for competing computational hypotheses. We evaluate the performance and interpretability of both approaches using an auditory neuroimaging dataset to reconstruct a known loudness-processing pathway. We discuss the implications of this shift for systems neuroscience, specifically regarding the handling of collinear models and the robust accumulation of evidence.

Many problems in science and engineering are difficult to model accurately, either due to unknown physical mechanisms, poorly quantified measurement uncertainty, or prohibitive computational costs of high-fidelity simulations. These challenges limit the applicability of classical probabilistic inference methods such as Markov chain Monte Carlo, especially in high-dimensional Bayesian inverse problems. As data from scientific experiments become increasingly available, machine learning methods offer a flexible alternative to explicit parametric modelling. We study neural likelihood approximation, where the goal is to learn the likelihood function directly from data without explicit knowledge of the underlying data-generating process. A common approach trains likelihood surrogates by minimizing the Kullback-Leibler divergence between the true posterior and an approximate posterior, which is equivalent to minimizing the expected negative log-likelihood. This work improves the theoretical foundations of neural likelihood approximation by alleviating limitations of restrictive model classes: we show that, by working with un-normalized potentials and folding normalization into the training objective, the resulting learning problem is strictly convex. We show that empirical minimizers of the resulting data-driven objective converge to the true likelihood as the sample size grows. Numerical experiments for the neural likelihood approximation are conducted for a deblurring and a non-linear PDE based imaging problem.

Collective emotion is often inferred from the tone of mass media, but such emotion is not directly observed. One approximation is to extract sentiment from text and use sentiment indexes as proxies to study the temporal organization of news sentiment. Using a daily index of U.S. economic news sentiment from 24 newspapers (1980-2025), we examine whether the response time of this sentiment process has changed. Although the average balance of positive and negative coverage has remained broadly stable, the persistence of news sentiment states has increased substantially. In dynamical terms, this implies longer residence times in optimistic or pessimistic regimes and weaker short-run correction of sentiment shocks. Complementary statistics show declining sentiment volatility, fewer reversals, and increasing bimodality, i.e. a stronger separation between positive and negative sentiment states. We also find an asymmetry between bursts of negative and positive sentiment, with negative bursts tending to last longer. These patterns are consistent with a minimal endogenous-memory model in which a slowly evolving latent sentiment component becomes more persistent while short-range corrective feedback weakens. The findings indicate a change in the temporal response of the U.S. economic newspaper sentiment index over the last 45 years, with sentiment shocks leaving longer traces than expected under short-memory exponential decay. News-based sentiment is thus better modeled as persistent episodes rather than as daily reactions that reset after each event.

In this paper, we study discrete quasi-copulas and discrete imprecise copulas of minimal range, which naturally correspond to alternating sign matrices. We identify dense alternating sign matrices as a particularly well-behaved subclass. We prove that the associated discrete imprecise copulas are always coherent and hence avoid sure loss. We further show that a particular extension of the corresponding discrete quasi-copulas to the full domain preserves this structure and yields self-dual imprecise copulas. To the best of our knowledge, this provides the first general result in the literature showing how properties established for discrete imprecise copulas can be preserved when extending them to their full-domain counterpart. We conclude the paper by showing that, for general alternating sign matrices, the associated discrete imprecise copulas need not avoid sure loss, further emphasizing the special role of the subclass of dense alternating sign matrices. We establish, however, that the class of discrete quasi-copulas of minimal range is invariant under all defect transformations

Weighted pairwise composite likelihoods based on nearest-neighbor (NN) pairs provide a scalable alternative to full likelihood inference for spatial random fields, but can remain expensive when moderately large NN neighborhoods are needed. We propose a stochastic acceleration that constructs the deterministic NN candidate graph and evaluates only a randomized subset of its pairwise contributions. We consider two thinning designs: Bernoulli thinning, which controls the retained-pair budget in expectation, and fixed-budget thinning, which enforces an exact budget through target-wise sampling without replacement. Simulation studies for Matérn covariance models suggest that, in the settings considered, retaining two pairs per observation provides a stable statistical–computational compromise. The stochastic NN pairwise estimators provide a faster alternative to a Vecchia-type Gaussian approximation when substantial reductions in covariance-fitting time are desired and a modest loss of efficiency is acceptable. This trade-off is especially favorable for the mean, scale, and sill parameters, while the main efficiency loss is concentrated on smoothness estimation. In an application to July average temperature over the western–central United States, based on 2.5 million WorldClim observations, the proposed estimators achieve predictive accuracy essentially indistinguishable from the Vecchia benchmark, with substantially shorter covariance-fitting time.

We develop the first statistical methodology for causal inference with video features as treatments. Video is the most engaging content modality on the internet. A central causal question is how audience reactions change in response to treatment features that unfold over the course of a video. Unfortunately, standard causal inference methods are not applicable because confounding features are latent, high-dimensional, and dynamically related to both the treatment sequence and the outcome trajectory. To address these challenges, we first reproduce each video using a deep generative model and leverage the model's internal representations as learned, low-dimensional summaries of video content for causal estimation. We then establish that the average potential-outcome trajectory under dynamic stochastic interventions is nonparametrically identified. Lastly, we propose a consistent and asymptotically normal estimator based on a longitudinal neural network architecture. We empirically validate our approach by constructing a new causal inference benchmark consisting of $10{,}000$ Super Mario Bros. levels played by fixed Mario AI agents, where ground-truth causal effects are known by construction. Finally, we apply our method to television advertisements from the 2020 U.S. presidential campaign and find that increasing the probability of a candidate appearing over time leads to higher average viewer evaluations. With the proposed methodology, researchers can ask which visual features, appearing at which points in a video, influence audience responses, while benchmarking new methods against datasets with known ground-truth causal effects.

We study binary classification under shared-generator elliptical class-conditional distributions. The log-likelihood ratio is an additive function of the two squared Mahalanobis radii, with radial link $\varphi=\log g$; QDA is recovered only when this link is affine. We derive the Bayes radial-link family from the within-class radius law and estimate it by a finite fractional-power stochastic-polynomial projection instead of tuning a generic spline. The link is identifiable from the radius law, the plug-in estimator is $\sqrt{n}$-consistent and asymptotically normal under finite-moment regularity conditions, and the induced classifier is asymptotically Bayes-optimal in an iterated sieve limit. The structural bridge, GAM membership, and identity-link/affine-generator dichotomy are verified in Lean 4 without unproven placeholders. Against the global Mahalanobis-GAM of Ghosh et al. (2025), reimplemented with mgcv REML splines at equal input budget, the derived link is never significantly worse on three UCI benchmarks and is decisively better on breast_cancer ($[+0.009,+0.021]$ global, $[+0.109,+0.136]$ global+local). Across six real financial series under temporal-dependence-robust validation, it is never significantly worse than the fitted GAM and is significantly better on three of five heavy-tailed series plus the light-tailed control. Relative to QDA, it improves the heaviest-tailed series (oil $[+0.024,+0.070]$, S&P 500 $[+0.038,+0.126]$, JPY/USD $[+0.009,+0.047]$) and ties elsewhere. A closed-form rate simulation corroborates the $\sqrt{n}$ rate and the predicted excess-risk dichotomy between QDA's approximation-limited floor and the derived link's vanishing excess risk. The contribution is no significant loss relative to a tuned global GAM without spline smoothing-parameter selection, plus improved accuracy over QDA where generator curvature matters.

We aim to identify scattering network architectures that maximize the separation capacity on data with low intrinsic dimension. The networks we consider employ a fixed monomial nonlinearity and no pooling, so that the only design variable is the frame generated by the network filters. For data modeled as rectifiable sets, we first characterize and bound the separation capacity of general feature extractors in terms of the geometry of the dataset. We then particularize to scattering networks and obtain two design criteria: (i) the filters should meet the data on sufficiently many frequencies, and (ii) the matrices coupling the frame to the geometry of the data should be well-conditioned.