Papers

In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks. We first show that any function in $W^{2,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy, measured in the $W^{1,\infty}$-norm, by a fixed-size neural network using the Elementary Universal Activation Function ($\mathrm{EUAF}$). To extend this result to $W^{s,\infty}((a,b)^d)$ for $s\in\mathbb{N}$, we introduce a smooth activation $\mathrm{DUAF}_{\infty}$ from the family of Differentiable Universal Activation Functions ($\mathrm{DUAF}_n$). We prove that any function in $W^{s,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy in the $W^{s-1,\infty}$-norm by a fixed-size $\mathrm{DUAF}_{\infty}$-activated network. We further construct sigmoidal variants $\widetilde{\mathrm{DUAF}}_n$ and show that, for every $1\leq s\leq n$, fixed-size $\widetilde{\mathrm{DUAF}}_n$-activated networks still approximate any $f\in W^{s,\infty}((a,b)^d)$ with arbitrary accuracy in the $W^{s-1,\infty}$-norm. In all these results, the width and depth bounds are computed explicitly, and the proposed activations are elementary.

The rapid adoption of generative AI and Large Language Models (LLMs) has spurred interest in synthetic data as a privacy-preserving alternative to sensitive real-world datasets. However, generating high-utility synthetic data often carries the risk of memorizing and regurgitating private information from the training corpus. In this work, we present a customizable empirical auditing framework designed to detect and explain such data disclosures. Our framework introduces a mechanism to distinguish between "true disclosures"-where the system directly reproduces a user's information-and "phantom disclosures”-where the system incidentally generates a user's data. By partitioning input data into training and holdout sets and applying rigorous statistical hypothesis testing, we determine if observed disclosures are consistent with strict privacy baselines, such as zero-learning or specific Differential Privacy (DP) bounds. Crucially, this approach requires no model access, no canary insertion, and no reference model training -only the synthetic output and a held-out control set. We demonstrate that this framework effectively functions as a membership inference attack, providing empirical lower bounds on privacy leakage that are tighter than prior data-based auditing methods. Our approach is model-agnostic, applies to any synthetic data generation mechanism, and requires orders of magnitude fewer computational resources than shadow-model or canary-based alternatives.

Detecting distributional differences between two independent samples is a fundamental problem in statistics and machine learning. Nonparametric two-sample testing provides a principled framework for determining whether two samples are drawn from the same underlying distribution, without assuming any specific parametric form for the distribution. In this study, we propose a new two-sample test statistic based on a newly introduced integral probability metric (IPM), using a specially designed parametric discriminator class with a single node of a neural network. We show that the resulting test statistic, called PReLU-IPM, is nonparametric and establish theoretical guarantees for the associated two-sample testing procedure, PReLU-TST, including its consistency and asymptotical equivalence to nonparametric IPM-based tests under regularity conditions. By analyzing multiple simulated and real benchmark datasets, we demonstrate that PReLU-TST achieves higher power across a range of alternatives or performs comparably to its competitors, for finite samples.

Functional optimization problems are typically solved by optimizing the parameters of a fixed representation, such as a neural network, resulting in highly nonconvex losses that complicate both training and theoretical analysis. An interesting alternative is functional gradient descent (FGD), that is, gradient descent directly in function space, which benefits from strong convergence results and admits a clean theory. However, FGD is difficult to implement in practice because functional gradients are infinite-dimensional, and thus cannot be fully computed nor stored in memory. Existing implementations therefore rely on fixed approximations, which introduce approximation error. We propose a new, theoretically-grounded FGD algorithm that adapts the representation of the functional gradients over the course of optimization. By explicitly incorporating this approximation into the analysis, we establish convergence to a stationary point (for smooth losses) and to a global minimizer (under smoothness + a Polyak-Lojasiewicz-type condition) regardless of our approximations. To the best of our knowledge, this is the first implementable FGD method with such guarantees in a general setting. We demonstrate the effectiveness of our method on regression, numerical solution of PDEs, and modern computer vision. Across settings, our method consistently outperforms both FGD with fixed approximations and neural network baselines in efficiency and accuracy.

Simulation-based inference (SBI) of latent parameters is often hindered by simulator misspecification, the mismatch between simulated and real-world observations caused by inherent modeling simplifications. RoPE, the recent state-of-the-art for robust SBI, addresses this through optimal transport between learned representations of real and simulated observations, but requires ground-truth parameter calibration pairs that are typically unavailable in the very settings where SBI is needed. What practitioners do have is unstructured side-information such as regime labels, instruction text, and policy bulletins. We propose Misspecification-Aware Simulation-Based Inference (MA-SBI), a calibration-free framework that turns this side-channel into a posterior correction. A learned corrector maps side-channel text to an observation-space shift applied before any pre-trained amortized posterior, requiring no retraining and no parameter ground-truth. Our main theorem bounds achievable bias reduction by the mutual information between misspecification and side-channel, with a non-vacuous constant that extends to all sub-Gaussian noise via Donsker-Varadhan. On hide-the-calibration benchmarks, MA-SBI with text alone matches the oracle posterior across 10 seeds and two backbones (TOST equivalence), while RoPE given more data does not. The two approaches are complementary: where misspecification is structural and recoverable from parameter pairs, RoPE dominates, as the theory predicts. A stochastic variant improves posterior-predictive log-likelihood on real COVID and OxCGRT epidemiological data, and correctly leaves the posterior unchanged on a well-specified cognitive-science corpus.

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.

Fairness metrics are essential for rigorously defining, quantifying, and mitigating biases in predictive models. While most existing metrics focus on binary classification tasks, fairness in time-to-event analyses has received limited attention. To address this gap, we propose a novel group fairness metric, the group-conditional Concordance Index (xCI), which extends Harrell's Concordance Index (CI) by conditioning on group membership. The xCI measures both within-group and cross-group ranking accuracy in the presence of right-censored data. We formally define the xCI, prove that CI is a weighted average of xCIs across all possible group pairs, and develop a consistent estimator using inverse probability of censoring weights (IPCW). We further investigate the relationship between xCI and predicted risk scores through analytical derivations and simulation studies. To demonstrate its practical utility, we present two case studies: (i) assessing the fairness of survival models trained on harmonized data from the Framingham Offspring, MESA, and ARIC studies, and (ii) evaluating fairness in existing cardiovascular disease (CVD) risk prediction models using Truveta, a large-scale electronic health record (EHR) database. Our results show that xCI effectively detects biases across demographic groups that are overlooked by existing metrics. Overall, xCI provides a valuable tool for fairness assessment in survival analysis, particularly in constrained resource allocation settings, and complements existing fairness evaluation approaches.

We propose a flexible framework for modeling the predictive distributions of nonlinear, possibly multivariate time series. Our approach expresses a general predictive distribution in an appropriate generative representation that is based on a folklore result from measure theoretic probability. This representation provides a direct simulation-based approximation to the predictive distribution, enabling straightforward computation of forecasts for the conditional mean and variance, fan charts, value at risk, expected shortfall, joint tail risks, and other quantities of interest. We estimate this generative representation using a version of conditional generative adversarial networks and provide a formal statistical analysis of estimation under weak temporal dependence. Specifically, estimation is expressed as a particular minimax problem and we establish consistency of its approximate solutions in Hausdorff distance. The empirical relevance of the approach is illustrated using applications to equity returns, realized variance, and realized covariances. The proposed method is also computationally manageable, with estimation in our applications taking approximately one minute on a standard laptop.

We re-interpret Transformer pretraining as a fast-slow, singularly perturbed flow along depth, with untied weights as its non-autonomous feature. The linearised dynamics is a depth-ordered product of layer maps. Along a token-homogeneous reference trajectory, the linearised layer factorises along the eigenbasis of a frozen attention kernel. Past a computable saturation depth, the flow factors through the block coarse-graining – in other words, running the layers is running the coarse variable, dually. Weight perturbations supported on the decaying bundle move neither the persistent component of the distinguished trajectory nor the frozen kernel to first order, so the framework partitions parameter space into visible and invisible directions, with the cross-block coupling of the slow path sitting entirely on the visible side. How large a gate the slow path can carry is bounded by a stability margin. On the data side: if block emissions follow an exponential family, block-mean pooling captures all the information the slow path can use; but if neighbouring blocks carry no shared structure, no cross-block channel can help the prediction, and the gate amplitude is invisible in the prediction risk. Stability delimits what the architecture may do; the data decides what it will.

Determining an appropriate sample size for a study is a crucial step in planning scientific research. Appropriate sample size planning avoids both inadequate and inflated sample sizes. Inflated sample sizes wastes resources, time and effort of human subjects, and lives of experimental animals. Inadequate sample sizes, a much more common problem, wastes even more resources through the inability to detect biologically meaningful differences and encourages questionable research practices like $p$-hacking. Microbiome studies are particularly challenged by small sample sizes, particularly in studies of human subjects or expensive animal models. In practice, the statistical power of taxa within a differential abundance study is influenced by the effect size (typically quantified as fold change), mean abundance of individual taxa, and the number of samples. We present a novel approach for sample size calculation for differential abundance studies as a function of effect size, mean abundance and statistical power of taxa. Our method is implemented in the power.nb R package, available at https://michaelagronah.com/power.nb/articles/stub.html. We applied our model for sample size calculation using estimates of mean abundance and fold change of taxa obtained from thirty real-world microbiome datasets. Our results showed that differential abundance microbiome studies require larger sample sizes than are currently prevalent in the literature to achieve adequate statistical power. Our framework will help researchers make informed decisions about appropriate sample sizes.

We study the problem of detecting a faint geometric signal hidden in an otherwise random graph. Formally, we consider a hypothesis testing problem in which, under the null, the observed graph is an Erdős–Rényi random graph $\mathcal{G}(n,q)$, while under the alternative a random geometric graph $\mathcal{G}(k,q,d)$ is planted on $k\le n$ vertices. The planted subgraph is generated from independent random points on the unit sphere $\mathbb{S}^{d-1}$, with edges determined by latent geometric proximity and calibrated to have edge density $q$. Our goal is to characterize the statistical and computational limits of detecting this hidden geometry. We derive sharp information-theoretic lower bounds that identify regimes where detection is impossible and provide algorithms that achieve these limits whenever detection is feasible. We further investigate the computational complexity of the problem and determine when efficient polynomial-time tests exist. The model exhibits an easy–hard–impossible phase transition: some regimes allow efficient detection, others permit detection only with computationally intractable procedures, and still others render detection impossible even with unlimited computational power. As evidence for the computational barrier, we prove that all low-degree polynomial algorithms fail throughout the conjecturally hard regime, demonstrating a sharp gap between statistical and computational feasibility.

Quantitative evidence synthesis method has become a central tool for integration of findings across multiple studies, multi-centre trials, and multi-source cohort data. However, the identification and interpretation of non-replicable, outlying, and influential studies remain insufficiently addressed in practice, despite their potential to substantially affect the robustness and credibility of meta-analytic conclusions. In this paper, we clarify the conceptual distinctions between non-replicability, statistical outlyingness, and study influence, emphasizing that these concepts are related but not interchangeable. We then review the standard principles and procedures of model diagnostics for detecting outlying and influential studies in meta-analysis, together with their underlying statistical rationale. Building on recent methodological developments, we further discuss several practical and methodological refinements, including approaches for handling imprecise and correlated sampling variances, robust diagnostic procedures, and graphical tools for facilitating the identification and interpretation of unusual studies. Finally, we summarize recent advances in outlier and influence diagnostics and provide recommendations for the cautious interpretation and evaluation of studies identified as potentially non-replicable, outlying, or influential within meta-analytic frameworks.

In process tracing, researchers ask how strongly their evidence favors their explanation, the working theory, over a rival. Fairfield and Charman (2022) compare the two theories with a Bayes factor whose probabilities are set by hand, which Zaks (2021) argues lets researchers overstate findings. We derive those probabilities from a fully specified model of the evidence a researcher could have examined, given which observations support each theory. The working theory says most of the evidence supports it. The rival says no more than half does. The model gives two Bayes factors, neither designed to favor the working theory. The second grants the rival more benefit of the doubt and is provably the smallest value the rival's claim allows. Researchers can report how much observation bias, re-coding, or a smoking-gun weight would change a conclusion. In six published studies we reanalyze, both Bayes factors exceed 20 for two studies and not for the other four.

Graph Laplacians encode graph structures in matrix form, and thus facilitate the application of linear algebra to graph theory. In statistics, two related families of probabilistic graphical models can be parameterized by graph Laplacians. The first one is the Laplacian-constrained Gaussian graphical model (LCGGM), which imposes that the (pseudo-)inverse covariance matrix of a Gaussian random vector is a Laplacian matrix. Applications include graph signal processing and network topology learning. The second one is the Hüsler-Reiss graphical model, which is considered as an extremal analog of the Gaussian graphical model, and can be used in extremal dependence modeling of floods, heatwaves, and financial losses. For both models, the restriction to positive edge weights in the graph Laplacian gives rise to an approach for graph structure learning that does not require tuning parameters. While these approaches yield a strong model fit in many settings, the resulting graph estimates are typically much denser than the underlying ground truth, limiting interpretability and scalability. In order to improve the accuracy of Laplacian-constrained graph learning, we propose to use spectral graph sparsification as a post-estimation operation. To do so, we replace the original Laplacian estimate by a sparser Laplacian that is spectrally close, and re-fit the model on the resulting graph. We refer to the two resulting methods as Spectral-LCGGM and Spectral-HR. We investigate the properties of the proposed estimators and show several theoretical results on their performance. Furthermore, we demonstrate that the newly proposed methods perform well by running simulations on Erdős-Rényi and stochastic block model graphs, and we also showcase their applications to real data.

We propose a Bayesian framework for joint distributional forecasting of monthly asylum applications across the EU-27. The model decomposes latent application intensities into country-specific random walks and common factors, with idiosyncratic and shared shocks allowed to exhibit heavy tails or stochastic volatility. Using Eurostat data from 2008 to 2026, we evaluate predictive distributions in a rolling out-of-sample exercise, scoring overall distributional accuracy and upper-tail risk. Three findings emerge. First, the preferred specification varies across countries, scoring rules, and horizons, underscoring the need to align models with policy-specific loss functions. Second, joint EU-27 models improve on country-by-country benchmarks, with the largest gains in the upper tail, where preparedness costs are most relevant. Third, random-walk log-intensities provide a useful short-run description of national asylum-application dynamics, especially when combined with flexible innovation dynamics. We conclude by discussing implications for national and EU-level agencies involved in asylum forecasting and preparedness planning.

Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood. In this work, we provide refined and novel convergence guarantees for Brownian motion based DFMs, focusing on the discretization error. Our analysis is conducted under the Kullback-Leibler (KL) divergence and the 2-Wasserstein distance. Under finite-moment conditions and a mild score integrability assumption, we derive KL convergence bounds with improved dimensional dependence compared to prior work, achieving, up to our knowledge, state-of-the-art scaling under minimal conditions. We further extend the analysis to the 2-Wasserstein distance: under an additional first-order score integrability assumption and a weak log-concavity condition, we obtain convergence guarantees with dimensional dependence consistent with the KL case.

Engineered robust losses such as Huber, Student-$t$, and generalised cross-entropy make supervised models tolerant of contamination but cannot answer which observations are corrupted. We introduce Neural Bayesian Anomaly Mitigation (NBAM), a general-purpose drop-in loss derived from a Bayesian latent-switch mixture model: the marginal likelihood defines a robust supervised loss, and the associated posterior defines an unsupervised contamination classifier. Like Huber or Student-$t$, NBAM can replace the standard training loss in any supervised pipeline; unlike them, it additionally learns a structured contamination model and returns a calibrated per-sample contamination posterior. A learned input-dependent prior $\pi_\phi(x)$ captures the spatial locality of contamination, so that samples near known corruptions are more likely to be flagged, while an Occam penalty emerges automatically and regularises against over-flagging. On CIFAR-10 with asymmetric label contamination, NBAM recovers the structure of the corruption process without supervision: the contamination posterior separates clean from corrupted samples, and the learned anomaly head identifies the direction of every label-flip pair. Alongside these capabilities, NBAM outperforms the four robust-loss baselines considered here at contamination rates 0.2-0.6.

The increasing demand for access to microdata in official statistics and data-intensive applications raises important challenges concerning disclosure risk, inferential validity and preservation of statistical utility. This paper proposes an interpretable energy-driven framework for privacy-aware synthetic data generation in mixed-type data. The proposed methodology combines discriminative modelling, Bayesian-Network proposal mechanisms, Metropolis–Hastings sampling and post-generation optimization within a constrained probabilistic framework. Unlike perturbation-based approaches, privacy-aware behaviour is achieved through constrained stochastic exploration guided by explicit plausibility, privacy, diversity and structural-coherence penalties. The framework is specifically designed for mixed-type tabular data characterized by sparse configurations, heterogeneous variable types and complex multivariate dependency structures. The generation process is formulated as a multi-objective sampling problem balancing statistical fidelity and disclosure-risk while preserving predictive utility. An extensive empirical evaluation is conducted using a mixed-type individual-level dataset containing demographic, behavioural and health-related variables. The validation strategy combines statistical fidelity diagnostics, predictive analyses, diversity measures, nearest-neighbour risk analysis, membership inference attacks and Split Conformal Prediction. The empirical results suggest that the proposed framework is capable of preserving a substantial portion of the predictive and multivariate structure of the original data while limiting exact memorization phenomena and maintaining favourable privacy-aware behaviour. The proposed methodology provides an interpretable framework for synthetic data generation under competing utility and privacy constraints.

We study Euler characteristics of scaled toric varieties arising from second hypersimplices. In algebraic statistics, these are closely connected to maximum likelihood (ML) degrees of toric models. We establish a correspondence between delta-matroids and the non-vanishing factors of the principal $A$-determinant, providing an explicit connection between delta-matroid theory and algebraic statistics. Using this framework, we show that a conjectured minimum ML degree is realizable by a suitable embedding of the variety. Furthermore, for second hypersimplices up to order six, we prove that this value is minimal among all embeddings, as conjectured by Clarke et al. (2024).

High-throughput gene expression data exhibit high dimensionality, complex intergene dependence, and pronounced biological heterogeneity across samples, presenting major challenges for unsupervised clustering and disease subtype discovery. We introduce a module-structured mixture factor model that combines finite mixture modeling with low-rank latent factor representations defined at the gene-module level. By explicitly modeling gene modules in both the mean and covariance structure, the proposed framework decomposes expression variability into global gene-specific effects, cluster-specific module-level shifts, latent dependence within modules, and gene-specific residual noise. An Expectation–Conditional Maximization algorithm is applied for parameter estimation, allowing stable and scalable inference in high-dimensional transcriptomic settings. This framework enables interpretable unsupervised identification of disease-associated molecular subtypes and phenotypic heterogeneity across two autoimmune diseases using a large clinical transcriptomic dataset.

The R package hyreg2 introduces a frequentist framework for estimating latent class models for mixed outcome types using a joint likelihood approach. The method combines continuous and dichotomous data under the assumption that both outcome types arise from a common underlying data-generating process. In the implemented model, continuous responses are assumed to follow a normal distribution, while dichotomous responses are modeled using a binomial distribution. Such models are used in various scientific disciplines to estimate a common set of parameters across different types of data (e.g. clinical trials, econometrics and health economics). Latent class estimation is performed using the expectation-maximization algorithm as implemented in the widely used R package flexmix. The hyreg2 package offers a user-friendly implementation of this joint likelihood framework, allowing users to estimate models without explicitly programming the likelihood function. Heteroskedasticity as well as censored data can be taken into account. In addition to model estimation, the package provides dedicated summary and visualization functions to facilitate the interpretation of results. The article presents the methodological framework underlying the package and illustrates its functionality through an example based on the estimation of an EQ-5D-5L value set.

Information criteria such as WAIC and WBIC extend model selection to singular learning machines, but they are usually derived for equilibrium posteriors. We formulate a finite-time analogue of WBIC by replacing the equilibrium posterior with an effective ensemble generated by learning dynamics under a resource constraint. When this ensemble admits an analytic effective potential, singular learning theory yields a resource-dependent real log canonical threshold. The resulting estimator gives a computable thermodynamic contribution to time-bounded MDL and identifies the finite-time singular complexity relevant to the structural information measured by epiplexity.

The Brody distribution, originally a phenomenological interpolation between Poisson and Wigner level-spacing statistics in quantum chaos, is calibrated here as a quantitative measure of short-range exclusion in 2D spatial point processes. Two results form the core. First, the 2D complete-spatial-randomness baseline is recalibrated to $\beta=0.96\pm0.15$, correcting the inappropriate 1D Poisson reference. Second, an empirical $\beta$–$r_{\text{excl}}$ calibration is validated against the effective hard-core radius with Spearman $\rho=0.988$. The framework is demonstrated on 58 manufactured surfaces (10 materials, 10 processes), phase-extracted interferometric profilometry of a certified roundness standard, and 2D binary embeddings of prime numbers. A sparse-integer control proves the prime $\beta=2.15$ signal is genuinely arithmetic ($\Delta\beta=+0.68$ over random-integer control), while a Cantor-embedding null result ($\beta=1.40$, TOST $p<0.01$) demonstrates that 2D exclusion is embedding-created rather than intrinsic. Density-thinning experiments establish that $\beta$ captures exclusion strength rather than point density, while absolute values are density-dependent. A distinct CSR baseline for binary fields at low fill fraction is identified, with a decision table provided. The $\beta$–$r_{\text{excl}}$ calibration, the CSR baseline correction, and the control protocols together constitute a calibrated measurement framework for reproducible characterisation of short-range exclusion in 2D spatial point processes.

Evaluating representation similarity is fundamental to representation learning. However, existing metrics suffer from significant limitations: they lack interpretability due to shifting baselines, lack robustness to outliers, and are computationally intractable for large datasets, forcing reliance on heuristic approximations. To address this, we develop an ordinal-similarity framework, instantiated by the Triplet (TSI) and Quadruplet (QSI) Similarity Indices, which measure alignment by quantifying the consistency of ordinal relationships. We theoretically demonstrate this formulation is inherently interpretable, robust to outliers, and computationally efficient. Finally, we establish a formal equivalence between TSI and local neighborhood alignment, measured by Mutual Nearest Neighbors. Empirically, we validate these properties and show that ordinal similarity offers a scalable approach to measuring alignment, enabling practitioners to better understand and design representations.

We study spectral concentration bounds for kernel matrices as approximation of the corresponding kernel integral operator. Results are established under weak assumptions on the data setting and the reproducing kernel relying only on a Mercer condition and a local Weyl law. This allows us to deal with key features of kernel matrices, such as large multiplicities, large effective dimension, and heavy-tailed distributions. Our results apply to infinite dimensional principal component analysis, manifold learning, and Bayesian nonparametric statistics. We illustrate this via two prototypical examples: The heat kernel on the sphere and a wavelet prior from Bayesian nonparametrics.

Domain Generalization (DG) aims to train models that generalize to unseen target domains but often overfit to domain-specific features, known as undesired correlations. Gradient-based DG methods typically guide gradients in a dominant direction but often inadvertently reinforce spurious correlations. Recent work has employed dropout to regularize overconfident parameters, but has not explicitly adjusted gradient alignment or ensured balanced parameter updates. We propose GENIE (Generalization-ENhancing Iterative Equalizer), a novel optimizer that leverages the One-Step Generalization Ratio (OSGR) to quantify each parameter's contribution to loss reduction and assess gradient alignment. By dynamically equalizing OSGR via a preconditioning factor, GENIE prevents a small subset of parameters from dominating optimization, thereby promoting domain-invariant feature learning. Theoretically, GENIE balances convergence contribution and gradient alignment among parameters, achieving higher OSGR while retaining SGD's convergence rate. Empirically, it outperforms existing optimizers and enhances performance when integrated with various DG and single-DG methods.

We give a characteristic-function formulation of Kunchenko's stochastic-polynomial construction for settings in which raw moments may fail to exist. In the finite-variance trigonometric case, the coefficients of the Kunchenko normal system are expressed through the characteristic function and its derivative. In the moment-free case, empirical characteristic functions on a fixed finite frequency grid define a bounded discrepancy geometry that remains meaningful for Cauchy, symmetric stable, and other heavy-tailed laws. We prove well-definedness and finite-grid almost sure consistency of this empirical characteristic-function geometry. We introduce the associated minimum-CF-distance estimator and establish its identifiability, strong consistency, and asymptotic normality on a fixed grid, with a covariance built from bounded trigonometric moments that stays finite even for Cauchy and stable laws; refining the grid increases the optimal-weight information monotonically to the Fisher information, so the estimator is asymptotically efficient in the dense-grid limit. We also relate bounded sine scores to weak stochastic-polynomial estimating equations. A small Lean 4 / Mathlib supplement checks selected deterministic identities underlying the bounded-score construction; convergence arguments and statistical interpretation remain outside the formalization.

We introduce, to our knowledge, the first deep generative modeling framework for probability distributions continuously supported on compact metric graphs. Given source and target measures on a metric graph, our method embeds the graph into a smooth ambient space, solves an entropic Kantorovich problem via a neural semidual parameterization, and projects generated samples back onto the original graph. We study two embedded geometries: an extrinsic Euclidean realization and the intrinsic tropical Abel–Jacobi embedding into the Jacobian torus. In both cases, the resulting generator is graph-supported by construction. We prove that, in the joint limit of increasing neural expressivity, the learned generator converges weakly to a valid transport coupling between the original graph measures. Empirically, across a range of geometrically distinct graphs, our method matches or improves upon heuristic transport baselines based on discrete graph OT, while scaling more favorably. Finally, we demonstrate scalability on real-world urban mobility data by training our model on one million Uber pickup locations in Manhattan, New York City.

Meta-analysis is a key statistical tool for synthesizing clinical trial data to evaluate treatment effects, yet traditional methods such as fixed-effect and random-effects models often fail to handle heterogeneity, study-level covariates, or hierarchical structures effectively. To overcome these limitations, we developed a Bayesian hierarchical meta-analysis framework for robust parameter estimation on small samples and utilized analytical integration for efficient inference. Simulation studies indicated robust estimation of the proposed model. We applied it to the efficacy and safety profiles of Oxcarbazepine (OXC) and Carbamazepine (CBZ) in epilepsy treatment. Gibbs sampling and analytical integration produced nearly identical results across all three clinical datasets. The results indicated no significant difference between OXC and CBZ in complete seizure control, while OXC was significantly associated with a higher probability of at least a 50% reduction in seizure frequency and a lower risk of side effects than CBZ. The code and relevant data used in this study are openly available on GitHub at https://github.com/xsjk/HierarchicalMetaAnalysis.

We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.