Papers

Towards understanding the statistical complexity of learning from heterogeneous sources, we study the problem of multi-distribution learning. Given $k$ data sources, the goal is to output a classifier for each source by exploiting shared structure to reduce sample complexity. We focus on the bounded label noise setting to determine whether the fast $1/\epsilon$ rates achievable in single-task learning extend to this regime with minimal dependence on $k$. Surprisingly, we show that this is not the case. We demonstrate that learning across $k$ distributions inherently incurs slow rates scaling with $k/\epsilon^2$, even under constant noise levels, unless each distribution is learned separately. A key technical contribution is a structured hypothesis-testing framework that captures the statistical cost of certifying near-optimality under bounded noise-a cost we show is unavoidable in the multi-distribution setting. Finally, we prove that when competing with the stronger benchmark of each distribution's optimal Bayes error, the sample complexity incurs a multiplicative penalty in $k$. This establishes a statistical separation between random classification noise and Massart noise, highlighting a fundamental barrier unique to learning from multiple sources.

Conditional independence tests (CIT) are widely used for causal discovery and feature selection. Even with false discovery rate (FDR) control procedures, they often fail to provide frequentist guarantees in practice. We highlight two common failure modes: (i) in small samples, asymptotic guarantees for many CITs can be inaccurate and even correctly specified models fail to estimate the noise levels and control the error, and (ii) when sample sizes are large but models are misspecified, unaccounted dependencies skew the test's behavior and fail to return uniform p-values under the null. We propose Empirically Calibrated Conditional Independence Tests (ECCIT), a method that measures and corrects for miscalibration. For a chosen base CIT (e.g., GCM, HRT), ECCIT optimizes an adversary that selects features and response functions to maximize a miscalibration metric. ECCIT then fits a monotone calibration map that adjusts the base-test p-values in proportion to the observed miscalibration. Across empirical benchmarks on synthetic and real data, ECCIT achieves valid FDR with higher power than existing calibration strategies while remaining test agnostic.

We study the use of exchangeable multi-task Gaussian processes (GPs) for causal inference in panel data, applying the framework to two settings: one with a single treated unit subject to a once-and-for-all treatment and another with multiple treated units and staggered treatment adoption. Our approach models the joint evolution of outcomes for treated and control units through a GP prior that ensures exchangeability across units while allowing for flexible nonlinear trends over time. The resulting posterior predictive distribution for the untreated potential outcomes of the treated unit provides a counterfactual path, from which we derive pointwise and cumulative treatment effects, along with credible intervals to quantify uncertainty. We implement several variations of the exchangeable GP model using different kernel functions. To assess prediction accuracy, we conduct a placebo-style validation within the pre-intervention window by selecting a “fake” intervention date. Ultimately, this study illustrates how exchangeable GPs serve as a flexible tool for policy evaluation in panel data settings and proposes a novel approach to staggered-adoption designs with a large number of treated and control units.

The group stage of a sports tournament is often made more appealing by introducing additional constraints in the group draw that promote an attractive and balanced group composition. For example, the number of intra-regional group matches is minimised in several World Cups. However, under such constraints, the traditional draw procedure may become non-uniform, meaning that the feasible allocations of the teams into groups are not equally likely to occur. Our paper quantifies this non-uniformity of the 2026 FIFA World Cup draw for the official draw procedure, as well as for 47 reasonable alternatives implied by all permutations of the four pots and two group labelling policies. We show why simulating with a recursive backtracking algorithm is intractable, and propose a workable implementation using integer programming. The official draw mechanism is found to be optimal based on four measures of non-uniformity. Nonetheless, non-uniformity can be more than halved if the organiser aims to treat the best teams drawn from the first pot equally.

Count data with an excessive number of zeros frequently arise in fields such as economics, medicine, and public health. Traditional count models often fail to adequately handle such data, especially when the relationship between the response and some predictors is nonlinear. To overcome these limitations, the partially linear zero-inflated Poisson (PLZIP) model has been proposed as a flexible alternative. However, all existing estimation approaches for this model are based on likelihood, which is known to be highly sensitive to outliers and slight deviations from the model assumptions. This article presents the first robust estimation method specifically developed for the PLZIP model. An Expectation-Maximization-like algorithm is used to take advantage of the mixture nature of the model and to address extreme observations in both the response and the covariates. Results of the algorithm convergence and the consistency of the estimators are proved. A simulation study under various contamination schemes showed the robustness and efficiency of the proposed estimators in finite samples, compared to classical estimators. Finally, the application of the methodology is illustrated through an example using real data.

Consider a $n\times n$ sparse non-Hermitian random matrix $X_n$ defined as the Hadamard product between a random matrix with centered independent and identically distributed entries and a sparse Bernoulli matrix with success probability $K_n/n$ where $K_n\le n$ (and possibly $K_n\ll n$) and $K_n\to \infty$ as $n\to \infty$. Let $E_n$ be a deterministic $n\times n$ finite-rank matrix. We prove that the outlier eigenvalues of $Y_n= X_n +E_n$ asymptotically match those of $E_n$. In the special case of a rank-one deformation, assuming further that the sparsity parameter satisfies $K_n \gg \log^9(n)$ and that the entries of the random matrix are sub-Gaussian, we describe the limiting behavior of the projection of the right eigenvector associated with the leading eigenvalue onto the right eigenvector of the rank-one deformation. In particular, we prove that the projection behaves as in the Hermitian case. To that end, we rely on the recent universality results of Brailovskaya and van Handel (2024) relating the singular value spectra of deformations of $X_n$ to Gaussian analogues of these matrices. Our analysis builds upon a recent framework introduced by Bordenave et.al. (2022), and amounts to showing the asymptotic equivalence between the reverse characteristic polynomial of the random matrix and a random analytic function on the unit disc with explicit dependence on the finite-rank deformation.

The paper outlines the principles of construction of a broad class of hierarchical aggregation algorithms of cluster analysis, essentially based on minimum distance mergers, which are derived from the general bi-partial objective function. It is shown how the algorithms arise from the bi-partial objective function, their affinity with the classical hierarchical aggregation algorithms is demonstrated, and the examples of such algorithms for the concrete forms of the bi-partial objective function are provided. This amounts to the first explicit and, at the same time, quite general, connection between optimization in clustering and the hierarchical aggregation algorithms. Thereby, the respective hierarchical algorithms gain a deeper justification, the means for evaluating the quality of clustering is provided, along with the criterion of stopping the cluster mergers.

Narratives about economic events and policies are widely recognised as influential drivers of economic and business behaviour. Yet the statistical identification of narrative emergence remains underdeveloped. Narratives evolve gradually, exhibit subtle shifts in content, and may exert influence disproportionate to their observable frequency, making it difficult to determine when observed changes reflect genuine structural shifts rather than routine variation in language use. We propose a statistical framework for detecting narrative emergence in longitudinal text corpora using Latent Dirichlet Allocation (LDA). We define emergence as a sustained increase in a topic's relative prominence over time and articulate a statistical framework for interpreting such trajectories, recognising that topic proportions are latent, model-estimated quantities. We illustrate the approach using a corpus of academic publications in economics spanning 1970-2018, where Nobel Prize-recognised contributions serve as externally observable signals of influential narratives. Topics associated with these contributions display sustained increases in estimated prevalence that coincide with periods of heightened citation activity and broader disciplinary recognition. These findings indicate that model-based topic trajectories can reflect identifiable shifts in economic discourse and provide a statistically grounded basis for analysing thematic change in longitudinal textual data.

This article presents a corrected version of the Satterthwaite (1941, 1946) approximation for the degrees of freedom of a weighted sum of independent variance components. The original formula is known to yield biased estimates when component degrees of freedom are small. The correction, derived from exact moment matching, adjusts for the bias by incorporating a factor that accounts for the estimation of fourth moments. We show that Kish's (1965) effective sample size formula emerges as a special case when all variance components are equal, and component degrees of freedom are ignored. Simulation studies demonstrate that the corrected estimator closely matches the expected degrees of freedom even for small component sizes, while the original Satterthwaite estimator exhibits substantial downward bias. Additional applications are discussed, including jackknife variance estimation, multiple imputation total variance, and the Welch test for unequal variances.

We propose a new test of uniformity on the hypersphere based on a Stein characterization associated with the Laplace-Beltrami operator. We identify a sufficient class of test functions for this characterization, linked to the moment generating function. Exploiting the operator's eigenfunctions to obtain a harmonic decomposition in terms of Gegenbauer polynomials, we show that the proposed procedure belongs to the class of Sobolev tests. We derive closed-form series representations for the asymptotic distribution of the test statistic under the null hypothesis and under fixed alternatives. To enhance power against a range of alternatives, we introduce a tuning parameter into the characterization and study its impact on rejection probabilities. We discuss data-driven strategies for selecting this parameter to maximize rejection rates for a given alternative and compare the resulting performance with that of related parametric tests. Additional numerical experiments compare the proposed test with competing Sobolev-class procedures, highlighting settings in which it offers clear advantages.

We introduce and develop a general paradigm for combining information across diverse data sources. In broad terms, suppose $\phi$ is a parameter of interest, built up via components $\psi_1,\ldots,\psi_k$ from data sources $1,\ldots,k$. The proposed scheme has three steps. First, the Independent Inspection (II) step amounts to investigating each separate data source, translating statistical information to a confidence distribution $C_j(\psi_j)$ for the relevant focus parameter $\psi_j$ associated with data source $j$. Second, Confidence Conversion (CC) techniques are used to translate the confidence distributions to confidence log-likelihood functions, say $\ell_{{\rm con},j}(\psi_j)$. Finally, the Focused Fusion (FF) step uses relevant and context-driven techniques to construct a confidence distribution for the primary focus parameter $\phi=\phi(\psi_1,\ldots,\psi_k)$, acting on the combined confidence log-likelihood. In traditional setups, the II-CC-FF strategy amounts to versions of meta-analysis, and turns out to be competitive against state-of-the-art methods. Its potential lies in applications to harder problems, however. Illustrations are presented, related to actual applications.

We introduce a new method for online parameter estimation in stochastic interacting particle systems, based on continuous observation of a small number of particles from the system. Our method recursively updates the model parameters using a stochastic approximation of the gradient of the asymptotic log likelihood, which is computed using the continuous stream of observations. Under suitable assumptions, we rigorously establish convergence of our method to the stationary points of the asymptotic log-likelihood of the interacting particle system. We consider asymptotics both in the limit as the time horizon $t\rightarrow\infty$, for a fixed and finite number of particles, and in the joint limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. Under additional assumptions on the asymptotic log-likelihood, we also establish an $\mathrm{L}^2$ convergence rate and a central limit theorem. Finally, we present several numerical examples of practical interest, including a model for systemic risk, a model of interacting FitzHugh–Nagumo neurons, and a Cucker–Smale flocking model. Our numerical results corroborate our theoretical results, and also suggest that our estimator is effective even in cases where the assumptions required for our theoretical analysis do not hold.

This paper develops a unified framework that links firm-level predictive signals, cross-asset spillovers, and the stochastic discount factor (SDF). Signals and spillovers are jointly estimated by maximizing the Sharpe ratio, yielding an interpretable SDF that both ranks characteristic relevance and uncovers the direction of predictive influence across assets. Out-of-sample, the SDF consistently outperforms self-predictive and expected-return benchmarks across investment universes and market states. The inferred information network highlights large, low-turnover firms as net transmitters. The framework offers a clear, economically grounded view of the informational architecture underlying cross-sectional return dynamics.

Stochastic network models play a central role across a wide range of scientific disciplines, and questions of statistical inference arise naturally in this context. In this paper we investigate goodness-of-fit and two-sample testing procedures for statistical networks based on the principle of maximum entropy (MaxEnt). Our approach formulates a constrained entropy-maximization problem on the space of networks, subject to prescribed structural constraints. The resulting test statistics are defined through the Lagrange multipliers associated with the constrained optimization problem, which, to our knowledge, is novel in the statistical networks literature. We establish consistency in the classical regime where the number of vertices is fixed. We then consider asymptotic regimes in which the graph size grows with the sample size, developing tests for both dense and sparse settings. In the dense case, we analyze exponential random graph models (ERGM) (including the Erdös-Rènyi models), while in the sparse regime our theory applies to Erdös-Rènyi graphs. Our analysis leverages recent advances in nonlinear large deviation theory for random graphs. We further show that the proposed Lagrange-multiplier framework connects naturally to classical score tests for constrained maximum likelihood estimation. The results provide a unified entropy-based framework for network model assessment across diverse growth regimes.

This is the guest editors' general introduction to a Special Issue of the Journal of Statistical Planning and Inference, dedicated to confidence distributions and related themes. Confidence distributions (CDs) are distributions for parameters of interest, constructed via a statistical model after analysing the data. As such they serve the same purpose for the frequentist statisticians as the posterior distributions for the Bayesians. There have been several attempts in the literature to put up a clear theory for such confidence distributions, from Fisher's fiducial inference and onwards. There are certain obstacles and difficulties involved in these attempts, both conceptually and operationally, which have contributed to the CDs being slow in entering statistical mainstream. Recently there is a renewed surge of interest in CDs and various related themes, however, reflected in both series of new methodological research, advanced applications to substantive sciences, and dissemination and communication via workshops and conferences. The present special issue of the JSPI is a collection of papers emanating from the Inference With Confidence workshop in Oslo, May 2015. Several of the papers appearing here were first presented at that workshop. The present collection includes however also new research papers from other scholars in the field.

The performance of kernel density estimators is usually studied via Taylor expansions and asymptotic approximation arguments, in which the bandwidth parameter tends to zero with increasing sample size. In contrast, this paper focusses directly on the finite-sample situation. Informative upper bounds are derived both for the integrated and the maximal mean squared error function. Results are reached for the traditional case, where the kernel is a probability density function, under various sets of assumptions on the underlying density to be estimated. Results are also derived for the important non-conventional case of the sinc kernel, which is not integrable and also takes negative values. We pin-point ways in which the sinc-based estimator performs better than the conventional kernel estimators. When proving our results we rely on methods related to characteristic and empirical characteristic functions.

Driven by the recent surge in neural-inspired modeling, point processes have gained significant traction in systems and control. While the Hawkes process is the standard model for characterizing random event sequences with memory, identifying its unknown kernels is often hindered by nonlinearity. Approaches using prescribed basis kernels have emerged to enable linear parameterization, yet they typically rely on iterative likelihood methods and lack rigorous analysis under model misspecification. This paper justifies a closed-form Least Squares identification framework for Hawkes processes with prescribed kernels. We guarantee estimator existence via the almost-sure positive definiteness of the empirical Gram matrix and prove convergence to the true parameters under correct specification, or to well-defined pseudo-true parameters under misspecification. Furthermore, we derive explicit Central Limit Theorems for both regimes, providing a complete and interpretable asymptotic theory. We demonstrate these theoretical findings through comparative numerical simulations.

Many causal quantities are only partially identifiable due to the inherent missingness of potential outcomes, and the associated partial identification (PI) sets can be obtained by solving an optimal transport (OT) problem. Covariates often provide additional information about the potential outcomes and thus yield tighter PI sets, which can be obtained via conditional optimal transport (COT). However, COT-based PI set estimators are susceptible to the curse of dimensionality in the covariates and outcomes, which precludes the asymptotic normality and hinders statistical inference. In this paper, we exploit smoothness in the marginal densities of covariates and potential outcomes and develop a wavelet-based primal method for COT with multivariate outcomes and covariates. Moreover, for quadratic cost functions, we establish a stability result for COT and prove asymptotic normality of the proposed estimator. This characterization of the asymptotic distribution enables valid statistical inference for the partial identification set. Empirically, we validate the estimation and inference performance of our approach through numerical experiments in comparison with existing benchmarks.

The recent developments of complex deep learning models have led to unprecedented ability to accurately predict across multiple data representation types. Conformal prediction for uncertainty quantification of these models has risen in popularity, providing adaptive, statistically-valid prediction sets. For classification tasks, conformal methods have typically focused on utilizing logit scores. For pre-trained models, however, this can result in inefficient, overly conservative set sizes when not calibrated towards the target task. We propose DANCE, a doubly locally adaptive nearest-neighbor based conformal algorithm combining two novel nonconformity scores directly using the data's embedded representation. DANCE first fits a task-adaptive kernel regression model from the embedding layer before using the learned kernel space to produce the final prediction sets for uncertainty quantification. We test against state-of-the-art local, task-adapted and zero-shot conformal baselines, demonstrating DANCE's superior blend of set size efficiency and robustness across various datasets.

In modern applications such as ECG monitoring, neuroimaging, wearable sensing, and industrial equipment diagnostics, complex and continuously structured data are ubiquitous, presenting both challenges and opportunities for functional data analysis. However, existing methods face a critical trade-off: conventional functional models are limited by linearity, whereas deep learning approaches lack interpretable region selection for sparse effects. To bridge these gaps, we propose a sparse Bayesian functional deep neural network (sBayFDNN). It learns adaptive functional embeddings through a deep Bayesian architecture to capture complex nonlinear relationships, while a structured prior enables interpretable, region-wise selection of influential domains with quantified uncertainty. Theoretically, we establish rigorous approximation error bounds, posterior consistency, and region selection consistency. These results provide the first theoretical guarantees for a Bayesian deep functional model, ensuring its reliability and statistical rigor. Empirically, comprehensive simulations and real-world studies confirm the effectiveness and superiority of sBayFDNN. Crucially, sBayFDNN excels in recognizing intricate dependencies for accurate predictions and more precisely identifies functionally meaningful regions, capabilities fundamentally beyond existing approaches.

We study stochastic gradient descent (SGD) for composite optimization problems with $N$ sequential operators subject to perturbations in both the forward and backward passes. Unlike classical analyses that treat gradient noise as additive and localized, perturbations to intermediate outputs and gradients cascade through the computational graph, compounding geometrically with the number of operators. We present the first comprehensive theoretical analysis of this setting. Specifically, we characterize how forward and backward perturbations propagate and amplify within a single gradient step, derive convergence guarantees for both general non-convex objectives and functions satisfying the Polyak–Łojasiewicz condition, and identify conditions under which perturbations do not deteriorate the asymptotic convergence order. As a byproduct, our analysis furnishes a theoretical explanation for the gradient spiking phenomenon widely observed in deep learning, precisely characterizing the conditions under which training recovers from spikes or diverges. Experiments on logistic regression with convex and non-convex regularization validate our theories, illustrating the predicted spike behavior and the asymmetric sensitivity to forward versus backward perturbations.

Functional covariates arise in many scientific and engineering applications when model inputs take the form of time-dependent or spatially distributed profiles, such as varying boundary conditions or changing material behaviours. In addition, new practices in digital simulation require predictions accompanied by confidence intervals. Models based on Gaussian processes (GPs) provide principled uncertainty quantification. However, GPs capable of jointly handling functional covariates and multiple correlated functional tasks remain largely under-explored. In this work, we extend the framework of GPs with functional covariates to multitask problems by introducing a fully separable kernel structure that captures dependencies across tasks and functional inputs. By taking advantage of the Kronecker structure of the covariance matrix, the model is made scalable. The proposed model is validated on a synthetic benchmark and applied to a realistic structure, a riveted assembly with functional descriptions of the material behaviour and response forces. The proposed functional multitask GP significantly improves over single task GPs. For the riveted assembly, it requires less than 100 samples to produce an accurate mean and confidence interval prediction. Despite its larger number of parameters, the multitask GP is computationally easier to learn than its single task pendant.

Mobile data technologies use “actigraphs” to furnish information on health variables as a function of a subject's movement. The advent of wearable devices and related technologies has propelled the creation of health databases consisting of human movement data to conduct research on mobility patterns and health outcomes. Statistical methods for analyzing high-resolution actigraph data depend on the specific inferential context, but the advent of Artificial Intelligence (AI) frameworks require that the methods be congruent to transfer learning and amortization. This article devises amortized Bayesian inference for actigraph time sheets. We pursue a Bayesian approach to ensure full propagation of uncertainty and its quantification using a hierarchical dynamic linear model. We build our analysis around actigraph data from the Physical Activity through Sustainable Transport Approaches in Los Angeles (PASTA-LA) study conducted by the Fielding School of Public Health in the University of California, Los Angeles. Apart from achieving probabilistic imputation of actigraph time sheets, we are also able to statistically learn about the time-varying impact of explanatory variables on the magnitude of acceleration (MAG) for a cohort of subjects.

Understanding minimal assumptions that enable learning and generalization is perhaps the central question of learning theory. Several celebrated results in statistical learning theory, such as the VC theorem and Littlestone's characterization of online learnability, establish conditions on the hypothesis class that allow for learning under independent data and adversarial data, respectively. Building upon recent work bridging these extremes, we study sequential decision making under distributional adversaries that can adaptively choose data-generating distributions from a fixed family $U$ and ask when such problems are learnable with sample complexity that behaves like the favorable independent case. We provide a near complete characterization of families $U$ that admit learnability in terms of a notion known as generalized smoothness i.e. a distribution family admits VC-dimension-dependent regret bounds for every finite-VC hypothesis class if and only if it is generalized smooth. Further, we give universal algorithms that achieve low regret under any generalized smooth adversary without explicit knowledge of $U$. Finally, when $U$ is known, we provide refined bounds in terms of a combinatorial parameter, the fragmentation number, that captures how many disjoint regions can carry nontrivial mass under $U$. These results provide a nearly complete understanding of learnability under distributional adversaries. In addition, building upon the surprising connection between online learning and differential privacy, we show that the generalized smoothness also characterizes private learnability under distributional constraints.

We study online maximization of non-monotone Diminishing-Return(DR)-submodular functions over down-closed convex sets, a regime where existing projection-free online methods suffer from suboptimal regret and limited feedback guarantees. Our main contribution is a new structural result showing that this class is $1/e$-linearizable under carefully designed exponential reparametrization, scaling parameter, and surrogate potential, enabling a reduction to online linear optimization. As a result, we obtain $O(T^{1/2})$ static regret with a single gradient query per round and unlock adaptive and dynamic regret guarantees, together with improved rates under semi-bandit, bandit, and zeroth-order feedback. Across all feedback models, our bounds strictly improve the state of the art.

We provide the first regression framework that simultaneously accommodates responses taking values in a general metric space and predictors lying on a general torus. We propose intrinsic local constant and local linear estimators that respect the underlying geometries of both the response and predictor spaces. Our local linear estimator is novel even in the case of scalar responses. We further establish their asymptotic properties, including consistency and convergence rates. Simulation studies, together with an application to real data, illustrate the superior performance of the proposed methodology.

Push-Sum-based decentralized learning enables optimization over directed communication networks, where information exchange may be asymmetric. While convergence properties of such methods are well understood, their finite-iteration stability and generalization behavior remain unclear due to structural bias induced by column-stochastic mixing and asymmetric error propagation. In this work, we develop a unified uniform-stability framework for the Stochastic Gradient Push (SGP) algorithm that captures the effect of directed topology. A key technical ingredient is an imbalance-aware consistency bound for Push-Sum, which controls consensus deviation through two quantities: the stationary distribution imbalance parameter $\delta$ and the spectral gap $(1-\lambda)$ governing mixing speed. This decomposition enables us to disentangle statistical effects from topology-induced bias. We establish finite-iteration stability and optimization guarantees for both convex objectives and non-convex objectives satisfying the Polyak–Łojasiewicz condition. For convex problems, SGP attains excess generalization error of order $\tilde{\mathcal{O}}\!\left(\frac{1}{\sqrt{mn}}+\frac{\gamma}{\delta(1-\lambda)}+\gamma\right)$ under step-size schedules, and we characterize the corresponding optimal early stopping time that minimizes this bound. For PŁ objectives, we obtain convex-like optimization and generalization rates with dominant dependence proportional to $\kappa\!\left(1+\frac{1}{\delta(1-\lambda)}\right)$, revealing a multiplicative coupling between problem conditioning and directed communication topology. Our analysis clarifies when Push-Sum correction is necessary compared with standard decentralized SGD and quantifies how imbalance and mixing jointly shape the best attainable learning performance.

The tremendous success of Transformer models in fields such as large language models and computer vision necessitates a rigorous theoretical investigation. To the best of our knowledge, this paper is the first work proving that standard Transformers can approximate Hölder functions $ C^{s,\lambda}\left([0,1]^{d\times n}\right) $$ (s\in\mathbb{N}_{\geq0},0<\lambda\leq1) $ under the $L^t$ distance ($t \in [1, \infty]$) with arbitrary precision. Building upon this approximation result, we demonstrate that standard Transformers achieve the minimax optimal rate in nonparametric regression for Hölder target functions. It is worth mentioning that, by introducing two metrics: the size tuple and the dimension vector, we provide a fine-grained characterization of Transformer structures, which facilitates future research on the generalization and optimization errors of Transformers with different structures. As intermediate results, we also derive the upper bounds for the Lipschitz constant of standard Transformers and their memorization capacity, which may be of independent interest. These findings provide theoretical justification for the powerful capabilities of Transformer models.

The choice of prior is central to solving ill-posed imaging inverse problems, making it essential to select one consistent with the measurements $y$ to avoid severe bias. In Bayesian inverse problems, this could be achieved by evaluating the model evidence $p(y \mid M)$ under different models $M$ that specify the prior and then selecting the one with the highest value. Diffusion models are the state-of-the-art approach to solving inverse problems with a data-driven prior; however, directly computing the model evidence with respect to a diffusion prior is intractable. Furthermore, most existing model evidence estimators require either many pointwise evaluations of the unnormalized prior density or an accurate clean prior score. We propose DiME, an estimator of the model evidence of a diffusion prior by integrating over the time-marginals of posterior sampling methods. Our method leverages the large amount of intermediate samples naturally obtained during the reverse diffusion sampling process to obtain an accurate estimation of the model evidence using only a handful of posterior samples (e.g., 20). We also demonstrate how to implement our estimator in tandem with recent diffusion posterior sampling methods. Empirically, our estimator matches the model evidence when it can be computed analytically, and it is able to both select the correct diffusion model prior and diagnose prior misfit under different highly ill-conditioned, non-linear inverse problems, including a real-world black hole imaging problem.

Incorporating external data can improve the efficiency of clinical trials, but distributional mismatches between current and external populations threaten the validity of inference. While numerous dynamic borrowing methods exist, the calibration of their borrowing parameters relies mainly on ad hoc, simulation-based tuning. To overcome this, we propose BOND (Borrowing under Optimal Nonparametric Distributional robustness), a framework that formalizes data noncommensurability through Wasserstein ambiguity sets centered at the current-trial distribution. By deriving sharp, closed-form bounds on the worst-case mean drift for both continuous and binary outcomes, we construct a distributionally robust, bias-corrected Wald statistic that ensures asymptotic type I error control uniformly over the ambiguity set. Importantly, BOND determines the optimal borrowing strength by maximizing a worst-case power proxy, converting heuristic parameter tuning into a transparent, analytically tractable optimization problem. Furthermore, we demonstrate that many prominent borrowing methods can be reparameterized via an effective borrowing weight, rendering our calibration framework broadly applicable. Simulation studies and a real-world clinical trial application confirm that BOND preserves the nominal size under unmeasured heterogeneity while achieving efficiency gains over standard borrowing methods.