We study harmonic map regression, a nonparametric estimator for manifold-valued responses, that penalizes the empirical Fréchet risk by the Dirichlet energy. By connecting penalized regression to the theory of harmonic maps, the estimator acquires a structural theory that parallels the classical Euclidean smoothing spline. The Euler-Lagrange equation characterizes the solution as a piecewise-geodesic spline, an equivalent kernel controls pointwise risk at the rate $n^{-2/3}$, and the infinite-dimensional variational problem reduces exactly to a finite-dimensional optimization. Such newly established connection reveals a topological phenomenon that has no analogue in Euclidean nonparametric regression and, to our knowledge, has not been studied in the manifold regression literature. On manifolds whose regression curves can wrap around in topologically distinct ways, maps in distinct homotopy classes are separated by energy barriers intrinsic to the geometry of the target, and the Dirichlet penalty makes the estimator sensitive to this structure, recovering the correct topological class with probability tending to one, a phase transition we call topological recovery. A curvature-dependent oracle inequality yields the minimax rate $n^{-2s/(2s+1)}$ for Sobolev order $s$, matching the Euclidean constant on non-positively curved targets, while five geometric obstructions show that the full structural theory is unique to the Dirichlet energy ($s=1$). Simulations on $S^2$, $\mathbb{H}^2$, $SO(3)$, $\mathrm{Sym}^+(2)$, and $T^2$ corroborate the theory, and an application to wind-direction data on $S^1$ illustrates practical advantages.

Multi-arm multi-stage (MAMS) trials have gained popularity, due to their improved efficiency in evaluating multiple treatments. A traditional MAMS trial often decreases the expected sample size of the trial compared to just running a multi-arm approach, but with the drawback of an increase in maximum sample size. For academic led trials this poses a particular challenge, as funding is typically based on the maximum required sample size. To address this, drop-the-loser designs were introduced, where a fixed number of treatments are dropped at each interim stage, thereby reducing the maximum sample size. In this work, we propose an enhanced multi-stage drop-the-loser design that also allows for early stopping of the entire trial for superiority. This approach aims to retain the benefits of a reduced maximum sample size while also lowering the expected sample size. The proposed design is motivated by a trial in atrial fibrillation. We derive analytical expressions for the type I error rate, power, and expected sample size, and compare the proposed design's performance to alternative methods. We outline the key requirements for implementing the proposed design and discuss the contexts in which it should be considered. For the motivating example the results show that the proposed design substantially reduces the expected sample size compared to a standard drop-the-loser design, while lowering the maximum sample size relative to running a traditional MAMS trial or multiple separate trials.

Comparing $K$-sample distributions is a fundamental problem in data science that arises in a wide variety of fields and applications. In this article, we introduce a maximum-of-differences approach to make such comparisons. Specifically, we first calculate the pairwise distances from the pooled observations of the $K$ samples. We then define the two observations as connected if their distance is less than a pre-specified threshold value. For each observation, we next calculate the ``within" and the ``between" probabilities associated with these two types of connections for the given observation, i.e., with other observations within the same sample and between the given observation and the observations in other samples. Subsequently, we propose a maximum-of-differences (MOD) test that finds the maximum value among the standardized squared differences between the ``within" and the ``between" probabilities of all observations. Accordingly, the proposed test is not only applicable to multivariate data with $K$ samples, but can also be extended to multivariate regression models. Furthermore, we obtain the covariance-adjusted (CA) version of the MOD (CA-MOD) test, which converges to the Type I extreme value distribution under some conditions. Moreover, we demonstrate the asymptotic properties of the two tests under both the null and alternative hypotheses. The performance and usefulness of the tests are illustrated via simulation studies and real examples.

Accurate representation of non-Gaussian distributions of quantities of interest in nonlinear dynamical systems is critical for estimation, control, and decision-making, but can be challenging when forward propagations are expensive to carry out. This paper presents an approach for estimating probability density functions of states evolving under nonlinear dynamics using Seminonparametric (SNP), or Gallant-Nychka, densities. SNP densities employ a probabilists' Hermite polynomial basis to model non-Gaussian behavior and are positive everywhere on the support by construction. We use Monte Carlo to approximate the expectation integrals that arise in the maximum likelihood estimation of SNP coefficients, and introduce a convex relaxation to generate effective initial estimates. The method is demonstrated on density and quantile estimation for the chaotic Lorenz system. The results demonstrate that the proposed method can accurately capture non-Gaussian density structure and compute quantiles using significantly fewer samples than raw Monte Carlo sampling.

Single-cell transcriptomic data approximates the abundance of proteins at a high resolution, but its noisiness necessitates transformation by a pipeline of methods before analysis and inference. In the absence of robust validation of these pipelines and methods, it remains unclear how best to process any particular dataset. To compensate for this, popular visualisation methods, e.g., t-SNE and UMAP, are commonly used to produce descriptions of datasets. Such visualisations are incomplete and provide subjective descriptions of samples rather than statistically meaningful statements about technical noise or biology. In this paper, we introduce the Zero-Inflated Negative-Binomial with Geometric Tail (ZINBGT), a mixture-model-based strategy for producing interpretable visualisations of each gene's expression across cells, along with diagnostic summaries that use Wasserstein distance to highlight outlier genes. These diagnostics are used to reveal an outlier gene within a T. brucei sample. This method is applied to a human immune-cell dataset, highlighting the relationship between sparsity, mean, and spread across genes, as well as revealing an issue with the use of zero-inflated negative-binomial distributions to model single-cell RNA data. An investigation of simulated datasets intended to replicate the immune-cell data revealed discrepancies with the ground truth, establishing purposes for which these simulated datasets are unsuitable. Finally, we list a number of different domains to which this method can be applied.

The Half-Trek Criterion (HTC) is the primary graphical tool for determining generic identifiability of causal effect coefficients in linear structural equation models (SEMs) with latent confounders. However, HTC is inherently node-wise: it simultaneously resolves all incoming edges of a node, leaving a gap of "inconclusive" causal effects (15-23% in moderate graphs). We introduce Iterative Identification Closure (IIC), a general framework that decouples causal identification into two phases: (1) a seed function S_0 that identifies an initial set of edges from any external source of information (instrumental variables, interventions, non-Gaussianity, prior knowledge, etc.); and (2) Reduced HTC propagation that iteratively substitutes known coefficients to reduce system dimension, enabling identification of edges that standard HTC cannot resolve. The core novelty is iterative identification propagation: newly identified edges feed back to unlock further identification -- a mechanism absent from all existing graphical criteria, which treat each edge (or node) in isolation. This propagation is non-trivial: coefficient substitution alters the covariance structure, and soundness requires proving that the modified Jacobian retains generic full rank -- a new theoretical result (Reduced HTC Theorem). We prove that IIC is sound, monotone, converges in O(|E|) iterations (empirically <=2), and strictly subsumes both HTC and ancestor decomposition. Exhaustive verification on all graphs with n<=5 (134,144 edges) confirms 100% precision (zero false positives); with combined seeds, IIC reduces the HTC gap by over 80%. The propagation gain is gamma~4x (2 seeds identifying ~3% of edges to 97.5% total identification), far exceeding gamma<=1.2x of prior methods that incorporate side information without iterative feedback.

We propose an adaptive MCMC method that learns a linear preconditioner which is dense in its off-diagonal elements but sparse in its parametrisation. Due to this sparsity, we achieve a per-iteration computational complexity of $O(m^2d)$ for a user-determined parameter $m$, compared with the $O(d^2)$ complexity of existing adaptive strategies that can capture correlation information from the target. Diagonal preconditioning has an $O(d)$ per-iteration complexity, but is known to fail in the case that the target distribution is highly correlated, see \citet[Section 3.5]{hird2025a}. Our preconditioner is constructed using eigeninformation from the target covariance which we infer using online principal components analysis on the MCMC chain. It is composed of a diagonal matrix and a product of carefully chosen reflection matrices. On various numerical tests we show that it outperforms diagonal preconditioning in terms of absolute performance, and that it outperforms traditional dense preconditioning and multiple diagonal plus low-rank alternatives in terms of time-normalised performance.

Statistical inference based on optimal transport offers a different perspective from that of maximum likelihood, and has increasingly gained attention in recent years. In this paper, we study univariate nonparametric shape-constrained density estimation via projection with respect to the $p$-Wasserstein distance, with a focus on the quadratic case $p = 2$. By considering shape constraints given by displacement convex subsets of the Wasserstein space, Wasserstein projection estimation is a convex optimization problem. We focus on two fundamental examples, namely non-increasing densities on $\mathbb{R}_+ := [0, \infty)$ and log-concave densities on $\mathbb{R}$. In each case, we prove structural properties of the Wasserstein projection estimator, propose a discretization which can be implemented by off-the-shelf solvers, and compare the projection estimator with the corresponding maximum likelihood estimator.

Watermarking for large language models (LLMs) has emerged as an effective tool for distinguishing AI-generated text from human-written content. Statistically, watermark schemes induce dependence between generated tokens and a pseudo-random sequence, reducing watermark detection to a hypothesis testing problem on independence. We develop a unified framework for LLM watermark detection based on e-processes, providing anytime-valid guarantees for online testing. We propose various methods to construct empirically adaptive e-processes that can enhance the detection power. The proposed methods are applicable to any sequential testing problem where independent pivotal statistics are available. In addition, theoretical results are established to characterize the power properties of the proposed procedures. Some experiments demonstrate that the proposed framework achieves competitive performance compared to existing watermark detection methods.

Two-sample testing, where we aim to determine whether two distributions are equal or not equal based on samples from each one, is challenging if we cannot place assumptions on the properties of the two distributions. In particular, certifying equality of distributions, or even providing a tight upper bound on the total variation (TV) distance between the distributions, is impossible to achieve in a distribution-free regime. In this work, we examine the blurred TV distance, a relaxation of TV distance that enables us to perform inference without assumptions on the distributions. We provide theoretical guarantees for distribution-free upper and lower bounds on the blurred TV distance, and examine its properties in high dimensions.

The opioid epidemic remains a major public health challenge in the United States, requiring a multi-pronged intervention approach to mitigate harms to communities. Given the heterogeneity of the epidemic across the country, it is crucial for policymakers to understand localized treatment effects of different intervention components and utilize limited resources efficiently. While locally calibrated simulation models offer a useful computational tool to project the epidemic outcomes for any given intervention policy, collecting simulation results for all intervention combinations to estimate localized treatment effects for each community is impractical because the number of possible intervention combinations grows exponentially with the number of interventions and levels at which they are applied. To tackle this, we develop a bi-level metamodel framework with a two-stage sequential design for efficient sampling. The metamodel consists of a response function linking health outcomes to each intervention component's treatment effect, and a Gaussian process regression to learn spatial and socio-economic structures of the treatment effects based on locally-contextualized covariates. With two-stage sequential sampling, we leverage spatial correlations and posterior uncertainty to sequentially sample the most informative counties and treatment conditions. We apply this framework to estimate treatment effects of buprenorphine dispensing and naloxone distribution on overdose mortality rates using a calibrated agent-based opioid epidemic model in PA counties. Our approach achieves approximately 5% average relative error using one-tenth the number of runs required for an exhaustive simulation. Our bi-level framework provides a computationally efficient approach to support policymakers, in evaluating resource-allocation strategies to mitigate the opioid epidemic in local communities.

A scale mixture of normals is a distribution formed by mixing a collection of normal distributions with fixed mean but different variances. A generalized gamma scale mixture draws the variances from a generalized gamma distribution. Generalized gamma scale mixtures of normals have been proposed as an attractive class of parametric priors for Bayesian inference in inverse imaging problems. Generalized gamma scale mixtures have two shape parameters, one that controls the behavior of the distribution about its mode, and the other that controls its tail decay. In this paper, we provide the first demonstration that the prior model is realistic for multiple large imaging data sets. We draw data from remote sensing, medical imaging, and image classification applications. We study the realism of the prior when applied to Fourier and wavelet (Haar and Gabor) transformations of the images, as well as to the coefficients produced by convolving the images against the filters used in the first layer of AlexNet, a popular convolutional neural network trained for image classification. We discuss data augmentation procedures that improve the fit of the model, procedures for identifying approximately exchangeable coefficients, and characterize the parameter regions that best describe the observed data sets. These regions are significantly broader than the region of primary focus in computational work. We show that this prior family provides a substantially better fit to each data set than any of the standard priors it contains. These include Gaussian, Laplace, $\ell_p$, and Student's $t$ priors. Finally, we identify cases where the prior is unrealistic and highlight characteristic features of images that suggest the model will fit poorly.

The practical use of future climate projections from global circulation models (GCMs) is often limited by their coarse spatial resolution, requiring downscaling to generate high-resolution data. Regional climate models (RCMs) provide this refinement, but are computationally expensive. To address this issue, machine learning (ML) models can learn the downscaling function, mapping coarse GCM outputs to high-resolution fields. Among these, generative approaches aim to capture the full conditional distribution of RCM data given coarse-scale GCM data, which is characterized by large variability and thus challenging to model accurately. We introduce EnScale, a generative ML framework emulating the full GCM-to-RCM map by training on multiple pairs of GCM and corresponding RCM data. It first adjusts large-scale mismatches between GCM and coarsened RCM data, followed by a super-resolution step to generate high-resolution fields. To efficiently model the high-dimensional output, the super-resolution step employs a novel class of sparse local stochastic layers. Both steps employ generative models optimized with the energy score, a proper scoring rule. Compared to state-of-the-art ML downscaling approaches, our setup reduces computational cost by about one order of magnitude. EnScale jointly emulates multiple variables -- temperature, precipitation, solar radiation, and wind -- spatially consistent over Central Europe. In addition, we propose a variant EnScale-t that enables temporally consistent downscaling. We establish a comprehensive evaluation framework across various categories including calibration, spatial and temporal structure, extremes, and multivariate dependencies. Comparison with diverse benchmarks demonstrates EnScale(-t)'s competitive performance and computational efficiency, offering a promising approach for accurate and temporally consistent RCM emulation.

Conformal prediction has emerged as a widely used framework for constructing valid prediction sets in classification and regression tasks. In this work, we extend the split conformal prediction framework to hierarchical classification, where prediction sets are commonly restricted to internal nodes of a predefined hierarchy, and propose two computationally efficient inference algorithms. The first algorithm returns internal nodes as prediction sets, while the second one relaxes this restriction. Using the notion of representation complexity, the latter yields smaller set sizes at the cost of a more general and combinatorial inference problem. Empirical evaluations on several benchmark datasets demonstrate the effectiveness of the proposed algorithms in achieving nominal coverage.

Let $N$ components be partitioned into two communities, denoted ${\cal P}_+$ and ${\cal P}_-$, possibly of different sizes. Assume that they are connected via a directed and weighted Erdös-Rényi (DWER) random graph with unknown parameter $ p \in (0, 1).$ The weights assigned to the existing connections are of mean-field-type, scaling as $N^{-1}$. At each time \modif{step}, we observe the state of each component: either it sends some signal to its successors (in the directed graph) or remains silent otherwise. In this paper, we show that it is possible to find the communities ${\cal P}_+$ and ${\cal P}_-$ based only on the activity of the $N$ components observed over $T$ time units. More specifically, we propose \modif{ two simple methods, an aggregated method and a spectral method, whose {\it misclassification rates} vanish as long as $T \gg N$ (up to log terms). This condition is proved to be near-optimal in the minimax sense. Moreover, under the stronger condition $T \gg N^2$ (up to log terms), the aggregated method is shown to achieve {\it exact recovery} with probability tending to $1$. } Interestingly, these simple \modif{methods} do not require any prior knowledge of the other model parameters (e.g. the edge probability $p$). The key step in our analysis is to derive an asymptotic approximation of the 1-lagged covariance matrix associated to the states of the $N$ components, as $N$ diverges. This asymptotic approximation relies on the study of the behavior of the solutions of a \modif{Stein-type} matrix equation satisfied by the simultaneous (0-lagged) covariance matrix associated to the states of the components. This study is challenging, especially because the simultaneous covariance matrix is random since it depends on the underlying DWER random graph.

Univariate marked Hawkes processes are used to model a range of real-world phenomena including earthquake aftershock sequences, contagious disease spread, content diffusion on social media platforms, and order book dynamics. This paper illustrates a fundamental connection between univariate marked Hawkes processes and multivariate Hawkes processes. Exploiting this connection renders a framework that can be built upon for expressive and flexible inference on diverse data. Specifically, multivariate unmarked Hawkes representations are introduced as a tool to parameterize univariate marked Hawkes processes. We show that such multivariate representations can asymptotically approximate a large class of univariate marked Hawkes processes, are stationary given the approximated process is stationary, and that resultant conditional intensity parameters are identifiable. A simulation study demonstrates the efficacy of this approach, and provides heuristic bounds for error induced by the relatively larger parameter space of multivariate Hawkes processes.

Most point process models for earthquakes currently in the literature assume the magnitude distribution is i.i.d. potentially hindering the ability of the model to describe the main features of data sets containing multiple earthquake mainshock aftershock sequences in succession. This study presents a novel multidimensional fractional Hawkes process model designed to capture magnitude dependent triggering behaviour by incorporating history dependence into the magnitude distribution. This is done by discretising the magnitude range into disjoint intervals and modelling events with magnitude in these ranges as the subprocesses of a mutually exciting Hawkes process using the Mittag-Leffler density as the kernel function. We demonstrate this model's use by applying it to two data sets, Japan and the Middle America Trench, both containing multiple mainshock aftershock sequences and compare it to the existing ETAS model by using information criteria, residual diagnostics and retrospective prediction performance. We find that for both data sets all metrics indicate that the multidimensional fractional Hawkes process performs favourably against the ETAS model. Furthermore, using the multidimensional fractional Hawkes process we are able to infer characteristics of the data sets that are consistent with results currently in the literature and that cannot be found by using the ETAS model.

Causality is pivotal to our understanding of the world, presenting itself in different forms: information-theoretic and relativistic, the former linked to the flow of information, the latter to the structure of space-time. Leveraging a framework introduced in PRA, 106, 032204 (2022), which formally connects these two notions in general physical theories, we study their interplay. Here, information-theoretic causality is defined through a causal modelling approach. First, we improve the characterization of information-theoretic signalling as defined through so-called affects relations. Specifically, we provide conditions for identifying redundancies in different parts of such a relation, introducing techniques for causal inference in unfaithful causal models (where the observable data does not "faithfully" reflect the causal dependences). In particular, this demonstrates the possibility of causal inference using the absence of signalling between certain nodes. Second, we define an order-theoretic property called conicality, showing that it is satisfied for light cones in Minkowski space-times with $d>1$ spatial dimensions but violated for $d=1$. Finally, we study the embedding of information-theoretic causal models in space-time without violating relativistic principles such as no superluminal signalling (NSS). In general, we observe that constraints imposed by NSS in a space-time and those imposed by purely information-theoretic causal inference behave differently. We then prove a correspondence between conical space-times and faithful causal models: in both cases, there emerges a parallel between these two types of constraints. This indicates a connection between informational and geometric notions of causality, and offers new insights for studying the relations between the principles of NSS and no causal loops in different space-time geometries and theories of information processing.

This paper extends the existing fractional Hawkes process to better model mainshock-aftershock sequences of earthquakes. The fractional Hawkes process is a self-exciting point process model with temporal decay kernel being a Mittag-Leffler function. A maximum likelihood estimation scheme is developed and its consistency is checked. It is then compared to the ETAS model on three earthquake sequences in Southern California. The fractional Hawkes process performs favourably against the ETAS model. Additionally, two parameters in the fractional Hawkes process may have a fixed geophysical meaning dependent on the study zone and the stage of the seismic cycle the zone is in.

This paper proposes new methodologies for conducting practical differentially private (DP) estimation and inference in high-dimensional linear regression. We first introduce a DP Bayesian Information Criterion (DP-BIC) for selecting the unknown sparsity parameter in differentially private sparse linear regression (DP-SLR), eliminating the need for prior knowledge of model sparsity, which is a requisite in the existing literature. Next, we develop the DP debiased algorithm that enables privacy-preserving inference on a particular subset of regression parameters. Our proposed method enables privacy-preserving inference on the regression parameters by leveraging the inherent sparsity of high-dimensional linear regression models. Additionally, we address private feature selection by considering multiple testing in high-dimensional linear regression by introducing a DP multiple testing procedure that controls the false discovery rate (FDR). This allows for accurate and privacy-preserving identification of significant predictors in the regression model. Through extensive simulations and real data analyses, we demonstrate the effectiveness of our proposed methods in conducting inference for high-dimensional linear models while safeguarding privacy and controlling the FDR.

We propose a Bayesian framework for planning simple step-stress accelerated life tests when items are subject to two independent competing failure modes We assume that the competing risks are independent, with lifetimes following Weibull distributions, and adopt the cumulative exposure model with a log-linear stress-life relationship to connect failure time distributions across stress levels. The optimality criterion is the preposterior variance of the $p$-th quantile of the lifetime distribution at use stress, evaluated without reliance on asymptotic approximations, making the methodology valid regardless of sample size. Building on the idea of quantile-based reparametrisation used in single-mode ALT \citep{zhang2006bayesian}, we extend this approach to the competing risks setting by reparametrising the model parameters for each failure mode to physically interpretable and approximately independent quantities, making it possible to elicit priors directly from engineering knowledge of device behaviour. Posterior inference is carried out using the No-U-Turn Sampler implemented in Stan, and the optimal design is located via Monte Carlo simulation over a grid of candidate designs. The methodology is illustrated on a real step-stress dataset for a solar lighting device subject to capacitor and controller failure modes. A comprehensive sensitivity analysis with respect to the quantile probability, the lower stress level, the prior hyperparameter specification, and the sample size shows that the optimal stress-change time is moderately sensitive to these inputs while the optimal lower stress level consistently favours operation close to use conditions, a finding that holds across all prior specifications considered.

We argue that Bonferroni correction is a better choice for online experimentation than it is commonly given credit for. The case rests on four considerations. First, it is the simplest broadly implementable FWER-controlling method that produces unconditional simultaneous confidence intervals for every metric. Second, in a well-specified decision framework, guardrail and quality metrics use intersection-union logic and cannot inflate the false positive rate, so the Bonferroni denominator is the number of success metrics only, not the total metric count. Third, it is uniquely tractable for pre-experiment sample size calculations. Fourth, we contextualise the power cost empirically. Drawing on a simulation study and an empirical analysis of 1,296 experiments run on Spotify's experimentation platform, Confidence, we show that the power loss relative to more sophisticated FWER methods depends on both how the correction family is specified and how many metrics are truly non-null. When guardrail metrics are incorrectly included in the family, Holm and Hommel are nearly indistinguishable from Bonferroni. When the family is correctly restricted to success metrics only, they gain roughly 4--5 percentage points in ship rate (the fraction of experiments where the treatment is deployed). When few metrics are truly non-null, the gap narrows to near zero regardless of method.

We develop a predictive-first optimisation framework for streaming hidden Markov models. Unlike classical approaches that prioritise full posterior recovery under a fully specified generative model, we assume access to regime-specific predictive models whose parameters are learned online while maintaining a fixed transition prior over regimes. Our objective is to sequentially identify latent regimes while maintaining accurate step-ahead predictive distributions. Because the number of possible regime paths grows exponentially, exact filtering is infeasible. We therefore formulate streaming inference as a constrained projection problem in predictive-distribution space: under a fixed hypothesis budget, we approximate the full posterior predictive by the forward-KL optimal mixture supported on $S$ paths. The solution is the renormalised top-$S$ posterior-weighted mixture, providing a principled derivation of beam search for HMMs. The resulting algorithm is fully recursive and deterministic, performing beam-style truncation with closed-form predictive updates and requiring neither EM nor sampling. Empirical comparisons against Online EM and Sequential Monte Carlo under matched computational budgets demonstrate competitive prequential performance.

We develop a theory of generalization and scaling for Mixture-of-Experts (MoE) Transformers that cleanly separates \emph{active} per-input capacity from routing combinatorics. By conditioning on fixed routing patterns and union-bounding across them, we derive a sup-norm covering-number bound whose metric entropy scales with the active parameter budget and incurs a MoE-specific routing overhead. Combined with a standard ERM analysis for squared loss, this yields a generalization bound under a $d$-dimensional manifold data model and $C^β$ targets, showing that approximation and estimation trade off as in dense networks once active parameters are accounted for appropriately. We further prove a constructive approximation theorem for MoE architectures, showing that, under the approximation construction, error can decrease either by scaling active capacity or by increasing the number of experts, depending on the dominant bottleneck. From these results we derive neural scaling laws for model size, data size, and compute-optimal tradeoffs. Overall, our results provide a transparent statistical reference point for reasoning about MoE scaling, clarifying which behaviors are certified by worst-case theory and which must arise from data-dependent routing structure or optimization dynamics.

This paper introduces a score-driven rating system, a generalization of the classical Elo rating system that employs the score, i.e. the gradient of the log-likelihood, as the updating mechanism for player and team ratings. The proposed framework extends beyond simple win/loss game outcomes and accommodates a wide range of game results, such as point differences, win/draw/loss outcomes, or complete rankings. Theoretical properties of the score are derived, showing that it has zero expected value, sums to zero across all players, and decreases with increasing value of a player's rating, thereby ensuring internal consistency and fairness. Furthermore, the score-driven rating system exhibits a reversion property, meaning that ratings tend to follow the underlying unobserved true skills over time. The proposed framework provides a theoretical rationale for existing dynamic models of sports performance and offers a systematic approach for constructing new ones.

Unobserved confounding is a key challenge when estimating causal effects from a treatment on an outcome in scientific applications. In this work, we assume that we observe a single, potentially multi-dimensional proxy variable of the unobserved confounder and that we know the mechanism that generates the proxy from the confounder. Under a completeness assumption on this mechanism, which we call Single Proxy Identifiability of Causal Effects or simply SPICE, we prove that causal effects are identifiable. We extend the proxy-based causal identifiability results by Kuroki and Pearl (2014); Pearl (2010) to higher dimensions, more flexible functional relationships and a broader class of distributions. Further, we develop a neural network based estimation framework, SPICE-Net, to estimate causal effects, which is applicable to both discrete and continuous treatments.

The most dangerous error in clinical trial interpretation is equating p > 0.05 with no effect. This review provides a practical, algorithm-based framework for classifying randomized controlled trial (RCT) results into six distinct categories positive, imprecise (+), neutral, inconclusive, negative, and harmful using confidence interval (CI) position relative to the minimal clinically important difference (MCID) as the primary tool, augmented by Bayesian posterior probabilities. We demonstrate that the same p > 0.05 result can represent three fundamentally different conclusions (inconclusive, negative, or neutral), show how Bayesian reanalysis can rescue benefit signals missed by frequentist thresholds, and illustrate the framework with real-world examples from critical care and cardiology trials. The framework synthesizes guidance from Altman, Harrell, Pocock, Zampieri, the ASA, and ICH E9 into a single coherent decision algorithm.

We study community detection in stochastic block models under pure node-level differential privacy, a stringent notion that protects the participation of an individual together with all of their incident edges. This setting is substantially more challenging than edge-private community detection, since modifying a single node can affect linearly many observations. On the algorithmic side, we analyze a node-private estimator based on the exponential mechanism combined with an extension lemma, and show that exact recovery remains achievable. In the standard sparse regime with logarithmic average degree and a fixed number of communities, our results imply that a logarithmic privacy budget suffices to obtain nontrivial recovery guarantees. On the lower bound side, we show that this logarithmic scaling is in fact unavoidable: any pure node-private method must fail to achieve polynomially small exact-recovery error, or polynomially small expected mismatch, unless the privacy budget is at least of this order. Moreover, in the regime of super-logarithmic privacy budgets, our upper and lower bounds yield a matching two-term characterization of the minimax risk, with one term governed by the non-private statistical signal and the other by the privacy budget; these match up to universal constants in the exponents. Taken together, our results identify an inherent logarithmic privacy cost in node-private community detection, absent under edge differential privacy, and provide a precise rate-level characterization of the tradeoff between node privacy and SBM recovery.

Constructing valid inferential methods for constrained parameters in normal and Poisson distributions represents two fundamental and important problems in applied statistics, for which there is currently no unified framework for statistical inference. Most existing studies assume that the nuisance parameters of the model are known, an assumption that is often impractical in real-world applications. However, under the more realistic scenario where nuisance parameters are unknown, the available Bayesian interval estimation methods fail to guarantee nominal coverage and thus cannot provide exact inference. To address these limitations, this paper develops prior-free inferential model (IM) approaches for parameters of interest in constrained normal and Poisson models and demonstrates that the confidence intervals (CIs) obtained from these novel IM methods can achieve exact nominal coverage. Furthermore, considering the discrete nature of the Poisson distribution, we employ random weighting techniques to improve the conservative coverage performance of the IM CIs. Simulation studies show that the coverage probabilities of the improved nonrandomized inferential model (NIM) CIs are closest to the prespecified nominal levels, with corresponding expected lengths shorter than those of Bayesian intervals in weak signal scenarios, whereas the shorter expected lengths of Bayesian intervals in strong signal scenarios come at the cost of sacrificing coverage guarantees. Therefore, the proposed IM and NIM CIs are superior to the Bayesian CIs. Finally, the advantages of the proposed methods are confirmed through an analysis of two experimental datasets on neutrinos in high-energy physics.

Although distributed lag non-linear models (DLNMs) are commonly used to quantify delayed and non-linear exposure-response relationships, most existing applications assume that these relationships are constant across space. However, in many geographical and environmental studies, local characteristics vary substantially across areas, making a spatially varying effect more realistic. Extending DLNMs to allow for spatial heterogeneity remains challenging, and only a limited number of modelling strategies have been proposed in literature. The most popular extension is a two-stage meta-analysis approach, which requires sufficiently large sample sizes at each location. Therefore, its usefulness is limited when working with sparse count data in small area data analyses. Although a number of alternative one-stage approaches have been introduced, their computational burden restricts their applicability in real-life data applications. In this paper, we introduce a computationally efficient Bayesian one-stage spatially-varying DLNM for count data. We define four model variants, differing in the assumed spatial dependence structure and the flexibility of the DLNM spline specification. To address the computational burden typically associated with these flexible models, we use Laplace approximations, offering an efficient alternative to classically used Markov Chain Monte Carlo (MCMC) approaches. Model comparison criteria are provided to facilitate the selection of a suitable model in a real-life data application. The proposed methods are evaluated through simulation studies, and their practical usefulness is illustrated through a real-life data application, investigating the temperature-mortality relationship in every municipality of Sicily, Italy.

Direct effect analyses usually require deciding whether a focal variable is a pre-exposure confounder or a post-exposure mediator. In observational studies, that distinction may be unclear because timing is measured coarsely or the variable reflects an evolving process. Considering the average treatment effect (ATE) and the natural direct effect (NDE) as a common notion of the direct effect when the focal variable is a confounder and a mediator, respectively, we show that, in general, no single observed-data estimand recovers both the ATE when the focal variable is a confounder and the NDE when it is a mediator. Consequently, if a practitioner applies an NDE estimator when the variable is actually pre-exposure, the resulting estimate may have no clear causal interpretation. We identify a no-additive-interaction condition under which these quantities coincide, develop sensitivity bounds for departures from that condition, and propose an alternative model-robust estimand. This estimand equals the ATE when the variable is pre-exposure and an interventional direct effect when it is post-exposure. Moreover, within a natural class of outcome-free stochastic direct effects, it is the unique observed-data functional that remains causally interpretable under both structural roles of the focal variable. We derive an efficient influence function and a doubly robust estimator, yielding robustness at two levels: the estimand is model-robust across the two causal scenarios, and the estimator is doubly robust with respect to nuisance estimation. In simulations and in an NHANES application on elevated PFAS burden, kidney function, and uric acid, mediation-based analyses yielded materially different reported estimates.

This paper introduces a projected functional gradient descent algorithm (P-FGD) for training nonparametric additive quantile regression models in online settings. This algorithm extends the functional stochastic gradient descent framework to the pinball loss. An advantage of P-FGD is that it does not need to store historical data while maintaining $O(J_t\ln J_t)$ computational complexity per step where $J_t$ denotes the number of basis functions. Besides, we only need $O(J_t)$ computational time for quantile function prediction at time $t$. These properties show that P-FGD is much better than the commonly used RKHS in online learning. By leveraging a novel Hilbert space projection identity, we also prove that the proposed online quantile function estimator (P-FGD) achieves the minimax optimal consistency rate $O(t^{-\frac{2s}{2s+1}})$ where $t$ is the current time and $s$ denotes the smoothness degree of the quantile function. Extensions to mini-batch learning are also established.

Causal discovery from observational data remains a fundamental challenge in machine learning and statistics, particularly when variables represent inherently positive quantities such as gene expression levels, asset prices, company revenues, or population counts, which often follow multiplicative rather than additive dynamics. We propose the Hybrid Moment-Ratio Scoring (H-MRS) algorithm, a novel method for learning directed acyclic graphs (DAGs) from positive-valued data by combining moment-based scoring with log-scale regression. The key idea is that for positive-valued variables, the moment ratio $\frac{\mathbb{E}[X_j^2]}{\mathbb{E}[(\mathbb{E}[X_j \mid S])^2]}$ provides an effective criterion for causal ordering, where $S$ denotes candidate parent sets. H-MRS integrates log-scale Ridge regression for moment-ratio estimation with a greedy ordering procedure based on raw-scale moment ratios, followed by Elastic Net-based parent selection to recover the final DAG structure. Experiments on synthetic log-linear data demonstrate competitive precision and recall. The proposed method is computationally efficient and naturally respects positivity constraints, making it suitable for applications in genomics and economics. These results suggest that combining log-scale modeling with raw-scale moment ratios provides a practical framework for causal discovery in positive-valued domains.

Randomized experiments have long been the gold standard for scientists seeking to learn about cause and effect. When randomized experiments are infeasible, scientists often resort to observational studies, which are widely available and often large but rely on untestable assumptions that, when violated, may result in biased estimates. Uncertainty about bias leads to a phenomenon known as the illusion of learning from observational research (Gerber, Green and Kaplan, 2004a): absent prior information about bias, observational results cannot meaningfully contribute to the estimation of a causal parameter. To shatter the illusion, we take an empirical Bayes perspective. We show that the distribution of observational biases can be learned from calibration studies-experiments that target a causal effect that is known a priori to be zero. Calibration identifies the distribution of observational bias and allows observational studies to inform the estimation of causal parameters via empirical Bayes shrinkage. We formalize the illusion phenomenon in an empirical Bayes setting and show that, with an increasing number of calibration and observation studies, both the bias distribution and the causal effect can be consistently recovered. We illustrate our method through a simulation study and a semi-synthetic application based on Ferraro and Miranda (2013)'s water-usage experiment.

The Hierarchical Kernel Transformer (HKT) is a multi-scale attention mechanism that processes sequences at L resolution levels via trainable causal downsampling, combining level-specific score matrices through learned convex weights. The total computational cost is bounded by 4/3 times that of standard attention, reaching 1.3125x for L = 3. Four theoretical results are established. (i) The hierarchical score matrix defines a positive semidefinite kernel under a sufficient condition on the symmetrised bilinear form (Proposition 3.1). (ii) The asymmetric score matrix decomposes uniquely into a symmetric part controlling reciprocal attention and an antisymmetric part controlling directional attention; HKT provides L independent such pairs across scales, one per resolution level (Propositions 3.5-3.6). (iii) The approximation error decomposes into three interpretable components with an explicit non-Gaussian correction and a geometric decay bound in L (Theorem 4.3, Proposition 4.4). (iv) HKT strictly subsumes single-head standard attention and causal convolution (Proposition 3.4). Experiments over 3 random seeds show consistent gains over retrained standard attention baselines: +4.77pp on synthetic ListOps (55.10+-0.29% vs 50.33+-0.12%, T = 512), +1.44pp on sequential CIFAR-10 (35.45+-0.09% vs 34.01+-0.19%, T = 1,024), and +7.47pp on IMDB character-level sentiment (70.19+-0.57% vs 62.72+-0.40%, T = 1,024), all at 1.31x overhead.

We study statistical parameter estimation in the setting of data markets. A buyer seeks to estimate a parameter based on samples that can be purchased from competing providers that differ in their data quality and provision costs. When quality is known ex ante, we define a cost-per-information score that summarizes each provider's provision cost per unit of information about the buyer's estimation objective. We describe second-score procurement mechanism that ranks providers by this score, and endogenously chooses both a provider and a sample size while making truthful cost reports optimal. We then turn to the more realistic setting where data quality is private, and can only be indirectly observed via the delivered data. In this setting, we propose a simple mechanism that augments the second-score rule with a lenient ex post statistical test of the reported quality. We prove that under mild conditions, there exists an equilibrium in which sellers report costs truthfully and report quality up to deviations that vanish as the procured sample size grows. Our analysis highlights how the choice of verification test and the buyer's accuracy-cost tradeoff jointly shape participation and misreporting incentives in data markets.

Digital firms routinely run many online experiments on shared user populations. When product decisions are compositional, such as combinations of interface elements, flows, messages, or incentives, the number of feasible interventions grows combinatorially, while available traffic remains limited. Overlapping experiments can therefore generate interaction effects that are poorly handled by decentralized A/B testing. We study how to design large-scale factorial experiments when the objective is not to estimate every treatment effect, but to identify a high-performing policy under a fixed experimentation budget. We propose a two-stage design that centralizes overlapping experiments into a single factorial problem and models expected outcomes as a low-rank tensor. In the first stage, the platform samples a subset of intervention combinations, uses tensor completion to infer performance on untested combinations, and eliminates weak factor levels using estimated marginal contributions. In the second stage, it applies sequential halving to the surviving combinations to select a final policy. We establish gap-independent simple-regret bounds and gap-dependent identification guarantees showing that the relevant complexity scales with the degrees of freedom of the low-rank tensor and the separation structure across factor levels, rather than the full factorial size. In an offline evaluation based on a product-bundling problem constructed from 100 million Taobao interactions, the proposed method substantially outperforms one-shot tensor completion and unstructured best-arm benchmarks, especially in low-budget and high-noise settings. These results show how centralized, policy-aware experimentation can make combinatorial product design operationally feasible at platform scale.

We study identification of a structural group effect when the group indicator $G\in\{0,1\}$ is unobserved but the analyst observes a calibrated probability score $p\in[0,1]$ satisfying $\mathbb{E}[G|p,X]=p$. Under a constant-coefficient structural mean model, the latent-group coefficient $τ$ is point-identified from the joint law of observables $(Y,X,p)$ by a simple ratio of weighted moments: the covariance of the signed score $2p-1$ with the covariate-partialled outcome, divided by twice the residual variance of the score after conditioning on covariates. Identification fails if and only if the score is a deterministic function of $X$; we establish this by constructing an explicit continuum of observationally equivalent models indexed by arbitrary values of $τ$. The identified coefficient differs from the marginal latent mean gap by a compositional term that is unidentified without further assumptions; we give a necessary and sufficient condition for the two to coincide. The oracle estimator is $\sqrt{n}$-consistent and asymptotically normal with a closed-form sandwich variance. Under calibration error bounded uniformly by $δ$, the bias is bounded by $|τ|\,\mathbb{E}[|2p-1|]\,δ\,(2V^*)^{-1}$, a bound that is sharp over all calibration error functions of that magnitude. Hard-threshold classification at $p=1/2$ attenuates the estimated gap by a factor strictly less than one. Monte Carlo experiments confirm the asymptotic theory, trace the divergence of RMSE as $V^*\to 0$, illustrate the attenuation bias of hard-threshold classification, and verify identification of the variance-weighted estimand under heterogeneous effects.

Computational models support high-stakes decisions across engineering and science, and practitioners increasingly seek probabilistic predictions to quantify uncertainty in such models. Existing approaches generate predictions either by sampling input parameter distributions or by augmenting deterministic outputs with uncertainty representations, including distribution-free and distributional methods. However, sampling-based methods are often computationally prohibitive for real-time applications, and many existing uncertainty representations either ignore input dependence or rely on restrictive Gaussian assumptions that fail to capture asymmetry and heavy-tailed behavior. Therefore, we extend the ACCurate and Reliable Uncertainty Estimate (ACCRUE) framework to learn input-dependent, non-Gaussian uncertainty distributions, specifically two-piece Gaussian and asymmetric Laplace forms, using a neural network trained with a loss function that balances predictive accuracy and reliability. Through synthetic and real-world experiments, we show that the proposed approach captures an input-dependent uncertainty structure and improves probabilistic forecasts relative to existing methods, while maintaining flexibility to model skewed and non-Gaussian errors.

Time-series stationarity is a property that statistical characteristics such as trend, variance, seasonality remain constant over time. It is considered fundamental to many forecasting and analysis methods. Different tests detect different types of non-stationarity: structural breaks or deterministic trends, clustered or time-dependent variance, stochastic or deterministic seasonality. A series might pass one test while failing another; single-test approaches seldom distinguish between conceptually different types of non-stationarity that require different types of tests and transformations. `StationarityToolkit` addresses this by providing a comprehensive Python library that runs 10 statistical tests across three categories: trend (4 tests), variance (4 tests), and seasonality (2 tests). Rather than a binary stationary/non-stationary verdict, users receive detailed diagnostics with actionable notes for each detection. The toolkit automatically infers the frequency of the data provided (requires datetime index), provides clear interpretations with test statistics and p-values, and supports an iterative test-transform-retest workflow essential for real-world data sets.

We develop a theoretical framework for generalization in the interpolating regime of statistical learning. The central question is why highly overparameterized estimators can attain zero empirical risk while still achieving nontrivial predictive accuracy, and how to characterize the boundary between benign and destructive overfitting. We introduce a spectral-transport stability framework in which excess risk is controlled jointly by the spectral geometry of the data distribution, the sensitivity of the learning rule under single-sample replacement, and the alignment structure of label noise. This leads to a scale-dependent Fredriksson index that combines effective dimension, transport stability, and noise alignment into a single complexity parameter for interpolating estimators. We prove finite-sample risk bounds, establish a sharp benign-overfitting criterion through the vanishing of the index along admissible spectral scales, and derive explicit phase-transition rates under polynomial spectral decay. For a model-specific specialization, we obtain an explicit theorem for polynomial-spectrum linear interpolation, together with a proof of the resulting rate. The framework also clarifies implicit regularization by showing how optimization dynamics can select interpolating solutions of minimal spectral-transport energy. These results connect algorithmic stability, double descent, benign overfitting, operator-theoretic learning theory, and implicit bias within a unified structural account of modern interpolation.

Most methods for learning with noisy labels require privileged knowledge such as noise transition matrices, clean subsets or pretrained feature extractors, resources typically unavailable when robustness is most needed. We propose Conformal Margin Risk Minimization (CMRM), a plug-and-play envelope framework that improves any classification loss under label noise by adding a single quantile-calibrated regularization term, with no privileged knowledge or training pipeline modification. CMRM measures the confidence margin between the observed label and competing labels, and thresholds it with a conformal quantile estimated per batch to focus training on high-margin samples while suppressing likely mislabeled ones. We derive a learning bound for CMRM under arbitrary label noise requiring only mild regularity of the margin distribution. Across five base methods and six benchmarks with synthetic and real-world noise, CMRM consistently improves accuracy (up to +3.39%), reduces conformal prediction set size (up to -20.44%) and does not hurt under 0% noise, showing that CMRM captures a method-agnostic uncertainty signal that existing mechanisms did not exploit.

This article proposes a biconvex modification to convex biclustering in order to improve its performance in high-dimensional settings. In contrast to heuristics that discard a subset of noisy features a priori, our method jointly learns and accordingly weighs informative features while discovering biclusters. Moreover, the method is adaptive to the data, and is accompanied by an efficient algorithm based on proximal alternating minimization, complete with detailed guidance on hyperparameter tuning and efficient solutions to optimization subproblems. These contributions are theoretically grounded; we establish finite-sample bounds on the objective function under sub-Gaussian errors, and generalize these guarantees to cases where input affinities need not be uniform. Extensive simulation results reveal our method consistently recovers underlying biclusters while weighing and selecting features appropriately, outperforming peer methods. An application to a gene microarray dataset of lymphoma samples recovers biclusters matching an underlying classification, while giving additional interpretation to the mRNA samples via the column groupings and fitted weights.

While clustering is ubiquitously used across science and industry, uncertainty in cluster assignments is rarely quantified with rigorous guarantees. We propose a novel conformal inference framework for clustering that returns confidence sets for cluster labels. The key challenge is that labels are unobserved and estimated from data, so naively using deterministic cluster labels can violate exchangeability and induce severe under-coverage. To address this, we propose split conformal clustering with stochastic labels, which samples labels from soft cluster labels, fits a soft classifier to predict these stochastic labels, and calibrates conformal scores to construct confidence sets for cluster labels at any query point. We establish a finite-sample lower bound on marginal coverage that reveals how under-coverage is controlled by two properties of the clustering algorithm: consistency of estimated soft labels and replace-one stability. Under mild conditions, we prove asymptotic coverage and verify these conditions for correctly specified parametric mixture models. Simulations for mixture models show that our method attains target coverage with informative set sizes, validating our theoretical results. Applications to clustering cell types in single-cell RNA-seq data demonstrate the practical utility and interpretability of our approach to quantifying cluster label uncertainty.

Attrition in survey and field experiments presents a challenge for social science research. Common approaches to deal with this problem -- such as complete case analysis, multiple imputation, and weighting methods -- rely on strong assumptions that may not hold in practice. This paper introduces a new method that combines recent advances in statistical inference with established tools for handling missing data. The approach produces prediction intervals for treatment effects that are both robust and precise. Evidence from simulation studies shows that the method achieves better coverage and produces narrower intervals than common alternatives. The reanalysis of two recently published experiment studies illustrates how this framework allows researchers to compare treatment effects across participants who remain in the study, those who drop out, and the full sample. Taken together, these results highlight how the proposed approach provides a stronger foundation for causal inference in the presence of attrition.

This paper studies the degree to which a bivariate copula fails to be symmetric under coordinate permutation, a property known as non-exchangeability. Working within an axiomatic framework that quantifies this asymmetry through a family of $L^p$-based measures, we establish sharp bounds linking non-exchangeability to classical dependence and concordance measures, prove exact scaling laws under convex mixing that enable explicit construction of copulas with any prescribed degree of asymmetry, and characterise the class of maximally non-exchangeable copulas together with the feasible range of asymmetry--concordance pairs. On the inferential side, we propose a nonparametric permutation test for exchangeability with exact finite-sample error control and consistency against all asymmetric alternatives, validated by Monte Carlo simulation and illustrated on a real data set.

Governments routinely adjust capacity in rationed programs such as university fields, medical training and public housing, where admitting one individual displaces others and triggers chains of reallocation. We show that in such settings, the standard multi-treatment two-stage least squares (2SLS) coefficient identifies exactly the total societal effect of a marginal expansion, including all downstream reallocations. The result is an algebraic identity: under instrument relevance and a single alignment condition, satisfied in centralized admissions systems, the 2SLS coefficient equals the general-equilibrium shadow value of relaxing a capacity constraint, while the single-instrument Wald ratio captures only the direct effect. Their difference recovers the full equilibrium adjustment without additional structure. Monotonicity is not required. The identity extends beyond queue-based allocation to any fixed-supply setting, including competitive markets with price instruments. We apply the framework to two policy questions in Swedish university admissions, where marginal students are allocated across fields through a centralized lottery mechanism. First, revisiting the debate on whether economics and business education erodes prosocial values, we find that the direct effect of expanding business on charitable giving is precisely zero, but expanding the less competitive fields that business students are displaced from has large prosocial effects. Second, analyzing gender-targeted STEM policies, we find that admitting four women to competitive STEM generates one additional male STEM degree through downstream vacancies. Both are general-equilibrium effects invisible to single-instrument methods.

While the Implicit Bias(or Implicit Regularization) of standard loss functions has been studied, the optimization geometry induced by discriminative metric-learning objectives remains largely unexplored.To the best of our knowledge, this paper presents an initial theoretical analysis of the implicit regularization induced by the Deep LDA,a scale invariant objective designed to minimize intraclass variance and maximize interclass distance. By analyzing the gradient flow of the loss on a L-layer diagonal linear network, we prove that under balanced initialization, the network architecture transforms standard additive gradient updates into multiplicative weight updates, which demonstrates an automatic conservation of the (2/L) quasi-norm.

Understanding how the composition of guest origin markets evolves over time is critical for destination marketing organizations, hospitality businesses, and tourism planners. We develop and apply Bayesian Dirichlet autoregressive moving average (BDARMA) models to forecast the compositional dynamics of guest origin market shares using proprietary Airbnb booking data spanning 2017--2025 across four major destination regions. Our analysis reveals substantial pandemic-induced structural breaks in origin composition, with heterogeneous recovery patterns across markets. In our analysis, the BDARMA framework achieves the lowest forecast error for EMEA and competitive performance across destination regions, outperforming standard benchmarks including naïve forecasts, exponential smoothing, and SARIMA on log-ratio transformed data in compositionally complex markets. For EMEA destinations, BDARMA achieves 27% lower forecast error than naïve methods ($p < 0.001$), with the greatest gains where multiple origin markets compete in the 5-25% share range. By modeling compositions directly on the simplex with a Dirichlet likelihood and incorporating seasonal variation in both mean and precision parameters, our approach produces coherent forecasts that respect the unit-sum constraint while capturing complex temporal dependencies. The methodology provides destination stakeholders with probabilistic forecasts of source market shares, enabling more informed strategic planning for marketing resource allocation, infrastructure investment, and crisis response.

We study information-theoretic phase transitions for the detectability of latent geometry in bipartite random geometric graphs RGGs with Gaussian d-dimensional latent vectors while only a subset of edges carries latent information determined by a random mask with i.i.d. Bern(q) entries. For any fixed edge density p in (0,1) we determine essentially tight thresholds for this problem as a function of d and q. Our results show that the detection problem is substantially easier if the mask is known upfront compared to the case where the mask is hidden. Our analysis is built upon a novel Fourier-analytic framework for bounding signed subgraph counts in Gaussian random geometric graphs that exploits cancellations which arise after approximating characteristic functions by an appropriate power series. The resulting bounds are applicable to much larger subgraphs than considered in previous work which enables tight information-theoretic bounds, while the bounds considered in previous works only lead to lower bounds from the lens of low-degree polynomials. As a consequence we identify the optimal information-theoretic thresholds and rule out computational-statistical gaps. Our bounds further improve upon the bounds on Fourier coefficients of random geometric graphs recently given by Bangachev and Bresler [STOC'24] in the dense, bipartite case. The techniques also extend to sparser and non-bipartite settings, at least if the considered subgraphs are sufficiently small. We furhter believe that they might help resolve open questions for related detection problems.

Metropolis algorithms are classical tools for sampling from target distributions, with broad applications in statistics and scientific computing. Their convergence speed is governed by the spectral gap of the associated Markov operator. Recently, Andrieu et al. (2024) derived the first explicit bounds for the spectral gap of Random--Walk Metropolis when the target distribution is smooth and strongly log-concave. However, existing literature rarely discusses non-smooth targets. In this work, we derive explicit spectral gap bounds for the random-walk Metropolis and Metropolis--adjusted Langevin algorithms over a broad class of non-smooth distributions. Moreover, combining our analysis with a recent result in Goyal et al. (2025), we extend these bounds to targets satisfying a Poincare or log-Sobolev inequality, beyond the strongly log-concave setting. Our theoretical results are further supported by numerical experiments.

We evaluate the misclustering probability of a spectral clustering algorithm under a Gaussian mixture model with a general covariance structure. The algorithm partitions the data into two groups based on the sign of the first principal component score. As a corollary of the main result, the clustering procedure is shown to be consistent in a high-dimensional regime.

This paper investigates symmetric composite binary quantum hypothesis testing (QHT), where the goal is to determine which of two uncertainty sets contains an unknown quantum state. While asymptotic error exponents for this problem are well-studied, the finite-sample regime remains poorly understood. We bridge this gap by characterizing the sample complexity -- the minimum number of state copies required to achieve a target error level. Specifically, we derive lower bounds that generalize the sample complexity of simple QHT and introduce new upper bounds for various uncertainty sets, including of both finite and infinite cardinalities. Notably, our upper and lower bounds match up to universal constants, providing a tight characterization of the sample complexity. Finally, we extend our analysis to the differentially private setting, establishing the sample complexity for privacy-preserving composite QHT.

Prevalence surveys are routinely used to monitor the effectiveness of mass drug administration (MDA) programmes for controlling neglected tropical diseases (NTDs). We propose a decay-adjusted spatio-temporal (DAST) model that explicitly accounts for the time-varying impact of MDA on NTD prevalence, providing a flexible and interpretable framework for estimating intervention effects from sparse survey data. Using case studies on soil-transmitted helminths and lymphatic filariasis, we show that DAST offers a practical alternative to standard geostatistical models when the objective includes quantifying MDA impact and supporting short-term programmatic forecasting. We also discuss extensions and identifiability challenges, advocating for data-driven parsimony over complexity in settings where the available data are too sparse to support the estimation of highly parameterised models.

Learning the structure of a Bayesian network from decentralized data poses two major challenges: (i) ensuring rigorous privacy guarantees for participants, and (ii) avoiding communication costs that scale poorly with dimensionality. In this work, we introduce Fed-Sparse-BNSL, a novel federated method for learning linear Gaussian Bayesian network structures that addresses both challenges. By combining differential privacy with greedy updates that target only a few relevant edges per participant, Fed-Sparse-BNSL efficiently uses the privacy budget while keeping communication costs low. Our careful algorithmic design preserves model identifiability and enables accurate structure estimation. Experiments on synthetic and real datasets demonstrate that Fed-Sparse-BNSL achieves utility close to non-private baselines while offering substantially stronger privacy and communication efficiency.

In many decision-making problems, the primary outcome is expensive, time-consuming, or difficult to observe, so individualized treatment rules (ITRs) may be instead learned from surrogate endpoints. However, a surrogate that is highly associated with the primary outcome, or even satisfies existing surrogate criteria, may not necessarily induce a treatment rule that performs well on the primary outcome, especially under treatment resource budget constraints. In this paper, we develop a principled framework for evaluating the decision-making value of surrogate endpoints. We introduce three ITR-oriented performance measures: surrogate regret, which assesses the expected loss from using the surrogate-optimal ITR instead of outcome-optimal ITR; surrogate gain, which quantifies the benefit of surrogate-optimal ITRs relative to the no-treatment baseline; and surrogate efficiency, which evaluates improvement over random treatment assignment. We also extend them to budget-constrained settings. We propose augmented inverse probability weighted (AIPW) estimators for these measures and establish their large-sample properties. We demonstrate the proposed approach on both simulations and an application to the Criteo dataset.

Efficacy testing is a cornerstone of clinical trials, ensuring that medical interventions achieve their intended therapeutic effects. Over the decades, a wide range of statistical methodologies have been developed to address the complexities of clinical trial data, including parametric, nonparametric, Bayesian, and machine learning approaches. Parametric methods, such as t-tests, ANOVA, and LMMs, have traditionally been the foundation of efficacy testing due to their efficiency under well-defined assumptions. Nonparametric techniques, including the Friedman test, Brunner-Munzel test, and modern extensions like nparLD, have emerged as robust alternatives, particularly for skewed, ordinal, or non-normal data. Bayesian methodologies have enabled the incorporation of prior information and uncertainty quantification, while machine learning techniques, such as deep learning and reinforcement learning, are revolutionizing trial designs and outcome predictions. Despite these advancements, significant gaps remain, including challenges in handling high-dimensional data, missingness, and ensuring equitable efficacy testing across diverse populations. This review provides a comprehensive overview of these statistical methods, highlighting their applications, strengths, limitations, and future directions. By bridging traditional statistical frameworks with modern computational techniques, the field can continue to advance toward more reliable and personalized clinical trial methodologies.

Inference in extreme value theory relies on a limited number of extreme observations, making estimation challenging. To address this limitation, we propose a non-parametric simulation scheme, the multivariate extreme events spectral bootstrap simulation procedure, relying on the spectral representation of multivariate generalized Pareto-distributed random vectors. Unlike standard bootstrap methods, our approach preserves the joint tail behaviour of the data and generates additional synthetic extreme data, thereby improving the reliability of inference. We demonstrate the effectiveness of our procedure on the estimation of tail risk metrics, under both simulated and real data. The results highlight the potential of this method for enhancing risk assessment in high-dimensional extreme scenarios.

Statistical learning methods for automated variable selection, such as the Least Absolute Shrinkage and Selection Operator (LASSO), elastic nets, and gradient boosting, have become increasingly popular tools for building powerful prediction models. Yet, in practice, analyses are often complicated by missing data. The most widely used approach to address missingness is multiple imputation, which involves creating several completed datasets. However, there is an ongoing debate about how to perform model selection in the presence of multiple imputed datasets. Simple strategies, such as pooling models across datasets, have been shown to have suboptimal properties. Although more sophisticated methods exist, they are often difficult to implement and therefore not widely applied. In contrast, two recent approaches extend the regularization methods LASSO and elastic nets to multiply imputed datasets by defining a single loss function, resulting in a unified set of coefficients across imputations. Our key contribution is to extend this principle to the framework of component-wise gradient boosting by proposing MIBoost, a novel algorithm that employs a uniform variable-selection mechanism across imputed datasets, together with its corresponding cross-validation routine MIBoostCV. In a simulation study, MIBoost yielded predictive performance comparable to that of other established methods, providing a practical boosting-based approach for variable selection with multiply imputed data. The proposed framework is implemented as the R package booami.

We present a novel Bayesian spatial disaggregation model for count data, providing fast and flexible inference at high resolution. First, it incorporates non-linear covariate effects using penalized splines, a flexible approach that is not typically included in existing spatial disaggregation methods. Additionally, it employs a spline-based low-rank kriging approximation for modeling spatial dependencies. The use of Laplace approximation provides computational advantages over traditional Markov Chain Monte Carlo (MCMC) approaches, facilitating scalability to large datasets. We explore two estimation strategies: one using the exact likelihood and another leveraging a spatially discrete approximation for enhanced computational efficiency. Simulation studies demonstrate that both methods perform well, with the approximate method offering significant computational gains. We illustrate the applicability of our model by disaggregating disease rates in the United Kingdom and Belgium, showcasing its potential for generating high-resolution risk maps. By combining flexibility in covariate modeling, computational efficiency and ease of implementation, our approach offers a practical and effective framework for spatial disaggregation.

We present `GL-LowPopArt`, a novel Catoni-style estimator for generalized low-rank trace regression. Building on `LowPopArt` (Jang et al., 2024), it employs a two-stage approach: nuclear norm regularization followed by matrix Catoni estimation. We establish state-of-the-art estimation error bounds, surpassing existing guarantees (Fan et al., 2019; Kang et al., 2022), and reveal a novel experimental design objective, $\mathrm{GL}(π)$. The key technical challenge is controlling bias from the nonlinear inverse link function, which we address with our two-stage approach. We prove a *local minimax lower bound*, showing that our `GL-LowPopArt` enjoys instance-wise optimality up to the condition number of the ground-truth Hessian. Our method immediately achieves an improved Frobenius error guarantee for generalized linear matrix completion. We also introduce a new problem setting called **bilinear dueling bandits**, a contextualized version of dueling bandits with a general preference model. Using an explore-then-commit approach with `GL-LowPopArt', we show an improved Borda regret bound over naïve vectorization (Wu et al., 2024).

Observational studies often present challenges for causal inference due to confounding and heterogeneity. In this paper, we illustrate how modern causal inference methods can be applied to large-scale academic salary data. Using records from 12,039 tenure-track faculty in the University of North Carolina system, linked with bibliometric indicators and institutional classifications, we estimate the causal effect of gender on faculty salaries. Our analysis combines propensity score matching with causal forests to adjust for rank, discipline, research productivity, and career experience. Results indicate that female faculty earn approximately 6% less than comparable male colleagues, with variation in the gap across career stages and levels of research productivity. This case study demonstrates how causal inference methods for observational data can provide insight into structural disparities in complex social systems.

We develop a new algorithm for inference in structural vector autoregressions (SVARs) identified with sign restrictions that can accommodate big data and modern identification schemes. The key innovation of our approach is to move beyond the traditional accept-reject framework commonly used in sign-identified SVARs. We show that an elliptical slice within Gibbs sampler can deliver dramatic gains in computational speed and render previously infeasible applications tractable. We also prove that the algorithm is well-defined, in the sense that its stationary distribution coincides with the posterior distribution of interest. To illustrate the approach in the context of sign-identified SVARs, we use a tractable example. We further assess the performance of our algorithm through two applications: a well-known small-SVAR model of the oil market featuring a tight identified set, and a large SVAR model with more than ten shocks and 100 sign restrictions.

Human aging is marked by a steady rise in the risk of dying with age-a process demographers call senescence. Over the past century, life expectancy has risen dramatically, but is this because we are aging slower, or simply starting it later? Vaupel hypothesizes that the pace at which individuals age may be constant, with gains in longevity coming from the delayed onset of senescence rather than its slowing down. We test this idea using a new framework that decomposes the pace of senescence into three components: a biological baseline, a long-term trend, and the cumulative impact of period shocks. Applying this to cohort mortality data above age 80 from 12 countries, we find that once period shocks are accounted for, there is no statistical evidence of a long-term trend, consistent with Vaupel's hypothesis. Analyses using lower starting ages yield the same qualitative conclusion. Rather than indicating a change in the process that drives senescence, these variations are consistent with echoes of shared historical events. These results suggest that while longevity has shifted, the rhythm of human aging may be conserved.

In interventional health studies, causal mediation analysis can be employed to investigate mechanisms through which the intervention affects the targeted health outcome. Identifying direct and indirect (i.e. mediated) effects from empirical data become complicated, however, when the mediator-outcome association is confounded by a variable itself affected by the treatment. Here, we investigate identification of mediational effects under such post-treatment confounding in a setting with a longitudinal mediator, time-to-event outcome and a trichotomous ordinal treatment-dependent confounder. If the intervention always affects the treatment-dependent confounder only in one direction (monotonicity), we show that the mediational effects are identified up to a stratum-specific sensitivity parameter and derive their empirical non-parametric expressions. The feasibility of the monotonicity assumption can be assessed using empirical data, based on restrictions on the marginal distributions of counterfactuals of the treatment-dependent confounder. We avoid pitfalls related to post-treatment conditioning by treating the mediator as a functional entity and defining the time-to-event outcome as a restricted disease-free time. In an empirical analysis, we use data from the Finnish Diabetes Prevention Study to assess the extent to which the effect of a lifestyle intervention on avoiding type 2 diabetes is mediated through weight reduction in a high-risk population, with other health-related changes acting as treatment-dependent confounders.

Scalability of Graph Neural Networks (GNNs) remains a significant challenge. To tackle this, methods like coarsening, condensation, and computation trees are used to train on a smaller graph, resulting in faster computation. Nonetheless, prior research has not adequately addressed the computational costs during the inference phase. This paper presents a novel approach to improve the scalability of GNNs by reducing computational burden during the inference phase using graph coarsening. We demonstrate two different methods -- Extra Nodes and Cluster Nodes. Our study extends the application of graph coarsening for graph-level tasks, including graph classification and graph regression. We conduct extensive experiments on multiple benchmark datasets to evaluate the performance of our approach. Our results show that the proposed method achieves orders of magnitude improvements in single-node inference time compared to traditional approaches. Furthermore, it significantly reduces memory consumption for node and graph classification and regression tasks, enabling efficient training and inference on low-resource devices where conventional methods are impractical. Notably, these computational advantages are achieved while maintaining competitive performance relative to baseline models.

Generative Flow Networks (GFlowNets) are amortized inference models designed to sample from unnormalized distributions over composable objects, with applications in generative modeling for tasks in fields such as causal discovery, NLP, and drug discovery. Traditionally, the training procedure for GFlowNets seeks to minimize the expected log-squared difference between a proposal (forward policy) and a target (backward policy) distribution, which enforces certain flow-matching conditions. While this training procedure is closely related to variational inference (VI), directly attempting standard Kullback-Leibler (KL) divergence minimization can lead to proven biased and potentially high-variance estimators. Therefore, we first review four divergence measures, namely, Renyi-$α$'s, Tsallis-$α$'s, reverse and forward KL's, and design statistically efficient estimators for their stochastic gradients in the context of training GFlowNets. Then, we verify that properly minimizing these divergences yields a provably correct and empirically effective training scheme, often leading to significantly faster convergence than previously proposed optimization. To achieve this, we design control variates based on the REINFORCE leave-one-out and score-matching estimators to reduce the variance of the learning objectives' gradients. Our work contributes by narrowing the gap between GFlowNets training and generalized variational approximations, paving the way for algorithmic ideas informed by the divergence minimization viewpoint.

Matrix recovery from sparse observations is an extensively studied topic emerging in various applications, such as recommendation system and signal processing, which includes the matrix completion and compressed sensing models as special cases. In this work we propose a general framework for dynamic matrix recovery of low-rank matrices that evolve smoothly over time. We start from the setting that the observations are independent across time, then extend to the setting that both the design matrix and noise possess certain temporal correlation via modified concentration inequalities. By pooling neighboring observations, we obtain sharp estimation error bounds of both settings, showing the influence of the underlying smoothness, the dependence and effective samples. We propose a dynamic fast iterative shrinkage thresholding algorithm that is computationally efficient, and characterize the interplay between algorithmic and statistical convergence. Simulated and real data examples are provided to support such findings.