Exponential families form the backbone of modern statistics and machine learning, but textbooks seldom derive them from first principles in an accessible way. Although minimal sufficiency and the principle of maximum entropy, originating in physics, provide core motivation, they are often presented as technical and requiring advanced prerequisites. Here, a short, self-contained derivation of exponential families based on maximum entropy is presented that is straightforward to carry out, requires only a modest background in information entropy, and avoids technicalities like constrained optimisation. Two propositions are demonstrated in this fashion: i) exponential families with a general base maximise information entropy with respect to that base subject to fixed expectations of canonical statistics, and ii) exponential families with a uniform base maximise standard information entropy under the same constraints. Maximum entropy therefore provides a principled foundation for exponential families with minimal prerequisites, highlighting the value of teaching entropy in statistics courses.

These lecture notes introduce the statistical analysis of continuous-time generative models built from Markov dynamics. We begin with the stochastic-calculus foundations of score-based diffusion models, including time reversal, score matching, and sampling from learned scores. We then present the broader framework of generator matching, which describes flows, diffusions, jump processes, and discrete generative models through their infinitesimal generators. We then focus on finite-sample guarantees. We explain how errors in the learned drift or generator propagate to the final generated distribution, why stability and regularity properties are essential, and how time-adaptive neural network classes can achieve optimal Wasserstein rates for smooth target distributions. Overall, the notes aim to connect modern generative modeling algorithms with the probabilistic, analytic, and statistical tools needed to understand their worst-case performance.

This study quantifies the association between air pollution and mortality in Ontario, Canada. Exposure-response relationships in air pollution epidemiology are complex due to three features: time-lagged associations, non-linear associations, and multiple pollutants. To address the first two features, two distinct classes of distributed lag non-linear model (DLNM) have been proposed, but extending them to multiple exposures and selecting an appropriate model remain challenging. We propose a unified framework for multiple exposure DLNMs, integrating model specification, estimation, selection and stacking. The framework applies to four different model structures: two additive and two proposed single-index DLNMs, all applicable to general outcome types, including the mortality counts in the motivating application. We develop an estimation approach that applies to all four models. Choosing among the candidate DLNMs is challenging a priori, and we derive an AIC to select among them. As an alternative to selecting a single model, we also extend a model stacking approach to combine inferences across the four DLNMs and propose an implementation scalable to our dataset with 106,346 observations. In the motivating analysis, the four DLNMs yield different estimates, and the proposed stacking approach identifies significant associations between respiratory mortality and a mixture of PM2.5, O3 and NO2.

We study the sharp bounds of $\mathbb{E}[X_1\cdots X_d]$ when the univariate marginal distributions are known, but the dependence structure between them is unspecified. Maximizing products over non-negative variables is straightforward via the comonotonic coupling, but the problem is more subtle when the marginals can take both positive and negative values. Specifically, two negative realizations can be matched to yield a positive product, whereas a single negative realization necessarily yields a negative product. We propose a decomposition of the problem into a magnitude part and a sign part, and show that universal upper and lower bounds for the product expectation follow from the comonotonic coupling of the absolute values and properly chosen sign vectors. Under a mild regularity assumption, we give necessary and sufficient conditions for these universal bounds to be attainable. For the upper bound, the marginal sign-bias vector must belong to the even-parity polytope, while for the lower, the corresponding condition involves the odd-parity polytope. We construct the extremal couplings via measurable selections on the parity polytope whenever these conditions hold. We study the case of identical marginals in more detail and provide examples of non-symmetric extremal coupling that achieve the universal bounds. We explicitly construct the extremal copulas in three dimensions, and use a recursive parity decomposition to obtain higher-dimensional extremal copulas from the trivariate ones.

Recovering latent structure from count data has received considerable attention in network inference, particularly when one seeks both cross-group interactions and within-group similarity patterns in bipartite networks, which is widely used in ecology research. Such networks are often sparse and inherently imperfect in their detection. Existing models mainly focus on interaction recovery, while the induced similarity graphs are much less studied. Moreover, sparsity is often not controlled, and scale is unbalanced, leading to oversparse or poorly rescaled estimates with degrading structural recovery. To address these issues, we propose a framework for structured sparse nonnegative low-rank factorization with detection probability estimation. We impose nonconvex $\ell_{1/2}$ regularization on the latent similarity and connectivity structures to promote sparsity within-group similarity and cross-group connectivity with better relative scale. The resulting optimization problem is nonconvex and nonsmooth. To solve it, we develop an ADMM-based algorithm with adaptive penalization and scale-aware initialization and establish its asymptotic feasibility and KKT stationarity of cluster points under mild regularity conditions. Experiments on synthetic and real-world ecological datasets demonstrate improved recovery of latent factors and similarity/connectivity structure relative to existing baselines.

The topic of Multivariate Time Series Anomaly Detection (MTSAD) has grown rapidly over the past years, with a steady rise in publications and Deep Learning (DL) models becoming the dominant paradigm. To address the lack of systematization in the field, this study introduces a novel and unified taxonomy with eleven dimensions over three parts (Input, Output and Model) for the categorization of DL-based MTSAD methods. The dimensions were established in a two-fold approach. First, they derived from a comprehensive analysis of methodological studies. Second, insights from review papers were incorporated. Furthermore, the proposed taxonomy was validated using an additional set of recent publications, providing a clear overview of methodological trends in MTSAD. Results reveal a convergence toward Transformer-based and reconstruction and prediction models, setting the foundation for emerging adaptive and generative trends. Building on and complementing existing surveys, this unified taxonomy is designed to accommodate future developments, allowing for new categories or dimensions to be added as the field progresses. This work thus consolidates fragmented knowledge in the field and provides a reference point for future research in MTSAD.

We prove tight lower bounds for online multicalibration, establishing an information-theoretic separation from marginal calibration. In the general setting where group functions can depend on both context and the learner's predictions, we prove an $Ω(T^{2/3})$ lower bound on expected multicalibration error using just three disjoint binary groups. This matches the upper bounds of Noarov et al. (2025) up to logarithmic factors and exceeds the $O(T^{2/3-\varepsilon})$ upper bound for marginal calibration (Dagan et al., 2025), thereby separating the two problems. We then turn to lower bounds for the more difficult case of group functions that may depend on context but not on the learner's predictions. In this case, we establish an $\widetildeΩ(T^{2/3})$ lower bound for online multicalibration via an $O(\log^3 T)$-sized group family constructed from an orthonormal basis, again matching upper bounds up to logarithmic factors.

We propose a new method for statistical inference in generalized linear models. In the overparameterized regime, Principal Component Regression (PCR) reduces variance by projecting high-dimensional data to a low-dimensional principal subspace before fitting. However, PCR incurs truncation bias whenever the true regression vector has mass outside the retained principal components (PC). To mitigate the bias, we propose Calibrated Principal Component Regression (CPCR), which first learns a low-variance prior in the PC subspace and then calibrates the model in the original feature space via a centered Tikhonov step. CPCR leverages cross-fitting and controls the truncation bias by softening PCR's hard cutoff. Theoretically, we calculate the out-of-sample risk in the random matrix regime, which shows that CPCR outperforms standard PCR when the regression signal has non-negligible components in low-variance directions. Empirically, CPCR consistently improves prediction across multiple overparameterized problems. The results highlight CPCR's stability and flexibility in modern overparameterized settings.

Recent causal inference literature has introduced causal effect decompositions to quantify sources of observed inequalities or disparities in outcomes, but these approaches are typically limited to pairwise comparisons. In healthcare delivery settings, both the exposure of interest-hospital or healthcare unit-and sociodemographic group membership may be polytomous, making pairwise contrasts inadequate. We therefore take the observed variance in care delivery outcomes as the quantity of interest and develop a new causal variance decomposition framework for this setting. The proposed framework attributes the observed variation to eight components, including novel terms characterizing modification of hospital effects by sociodemographic group membership, hospital access or selection, and the correlation between these two sources of heterogeneity. We discuss the causal interpretation of these components, propose both parametric and nonparametric model-based estimators, and study their performance through simulation. Finally, we illustrate the method using data from the SEER program in an application to cervical cancer care delivery.

We study the problem of modelling high-dimensional, heavy-tailed time series data via a factor-adjusted vector autoregressive (VAR) model, which simultaneously accounts for pervasive co-movements of the variables by a handful of factors, as well as their remaining interconnectedness using a sparse VAR model. To handle heavy tails, we propose an element-wise data truncation step followed by a two-stage estimation procedure for estimating the latent factors and the VAR parameter matrices. Assuming the existence of the $(2 + 2ε)$-th moment only for some $ε\in (0, 1)$, we derive the rates of estimation which, making explicit the effect of heavy tails through $ε$, are comparable to the rates attainable in light-tailed settings as $ε\to 1$. Numerically, we demonstrate the competitive performance of the proposed estimators on simulated datasets and in an application to forecasting macroeconomics indicators.

Identifying dependency between two random variables is a fundamental problem. The clear interpretability and ability of a procedure to provide information on the form of possible dependence is particularly important when exploring dependencies. In this paper, we introduce a novel method that employs a new estimator of the quantile dependence function and pertinent local acceptance regions. This leads to an insightful visualisation and a rigorous evaluation of the underlying dependence structure. We also propose a test of independence of two random variables, pertinent to this new estimator. Our procedures are based on ranks, and we derive a finite-sample theory that guarantees the inferential validity of our solutions at any given sample size. The procedures are simple to implement and computationally efficient. The large sample consistency of the proposed test is also proved. We show that, in terms of power, the new test is one of the best statistics for independence testing when considering a wide range of alternative models. Finally, we demonstrate the use of our approach to visualise dependence structure and to detect local departures from independence through analysing some real-world datasets.

Predicting customers' long-term revenue from sparse and irregular transaction data is central to marketing resource allocation in non-contractual settings, yet existing approaches face a trade-off. Traditional probabilistic customer base models deliver robust long-horizon forecasts by imposing strong structural assumptions, while flexible machine-learning models often require substantial training data and careful tuning. We propose a variational-autoencoder-based model that preserves the process-based likelihood of established attrition-transaction-spend models conditional on customer heterogeneity, but replaces the restrictive parametric mixing distribution with a flexible latent representation learned by encoder-decoder networks. The resulting approach (i) provides a single model for customer attrition, transactions and spending, (ii) remains reliable when contextual covariates are unavailable, and (iii) flexibly incorporates rich covariates and nonlinear effects when they are available. This design balances structural stability with the flexibility needed to capture complex purchase dynamics. Across multiple real-world datasets and prediction horizons, the proposed model improves upon the latest benchmarks. Businesses benefit directly, as a better assessment of customers' future revenues improves the efficiency of campaign targeting. For research, this work provides guidance on how to embed domain-specific models into the variational autoencoder framework, enabling flexible representation learning while retaining an econometrically meaningful process structure.

The Gaussian mixture model is widely used in unsupervised learning, owing to its simplicity and interpretability. However, a fundamental limitation of the classical Gaussian mixture model is that it forces each observation to belong to exactly one component. In many practical applications, such as genetics, social network analysis, and text mining, an observation may naturally belong to multiple components or exhibit partial membership in several latent components. To overcome this limitation, we propose the mixed membership sub-Gaussian model, which extends the classical Gaussian mixture framework by allowing each observation to belong to multiple components. This model inherits the interpretability of the classical Gaussian mixture model while offering greater flexibility for capturing complex overlapping structures. We develop an efficient spectral algorithm to estimate the mixed membership of each individual observation, and under mild separation conditions on the component centres, we prove that the estimation error of the per-individual membership vector can be made arbitrarily small with high probability. To our knowledge, this is the first work to provide a computationally efficient estimator with such a vanishing-error guarantee for a mixed-membership extension of the Gaussian mixture model. Extensive experimental studies demonstrate that our method outperforms existing approaches that ignore mixed memberships.

As the demand to integrate Artificial Intelligence into high-stakes environments continues to grow, explaining the reasoning behind neural-network predictions has shifted from a theoretical curiosity to a strict operational requirement. Our work is motivated by the explanations of autoregressive neural predictions on dynamic physical fields, as in weather forecasting. Gradient-based feature attribution methods are widely used to explain the predictions on such data, in particular due to their scalability to high-dimensional inputs. It is also interesting to remark that gradient-based techniques such as SmoothGrad are now standard on images to robustify the explanations using pointwise averages of the attribution maps obtained from several noised inputs. Our goal is to efficiently adapt this aggregation strategy to dynamic physical fields. To do so, our first contribution is to identify a fundamental failure mode when averaging perturbed attribution maps on dynamic physical fields: stochastic input perturbations do not induce stationary amplitude noise in attribution maps, but instead cause a geometric displacement of the attributions. Consequently, pointwise averaging blurs these spatially misaligned features. To tackle this issue, we introduce WassersteinGrad, which extracts a geometric consensus of perturbed attribution maps by computing their entropic Wasserstein barycenter. The results, obtained on regional weather data and a meteorologist-validated neural model, demonstrate promising explainability properties of WassersteinGrad over gradient-based baselines across both single-step and autoregressive forecasting settings.

Problem definition: Data-driven models in machine learning have enabled efficient management of production systems. However, a majority of machine learning models are devoted to modeling the mean response or average pattern, which is inappropriate for studying abnormal extreme events that are often of primary interest in aircraft manufacturing. Since extreme events from heavy-tailed distributions give rise to prohibitive expenditures in system management, sophisticated extreme models are urgently needed to analyze complex extreme risks. Engineering applications of extreme models usually focus on individual extreme events, which is insufficient for complex systems with correlations. Methodology/results: We introduce an extreme spatial model for multi-output response control systems that efficiently captures the dynamics using a bilinear function on two spatial domains for control variables and measurement locations. Marginal parameter modeling and extremal dependence have been investigated. In addition, an efficient graph-assisted composite likelihood estimation and corresponding computational algorithms are developed to cope with high-dimensional outputs. The application to composite aircraft production shows that the proposed model enables comprehensive analyses with superior predictive performance on extreme events compared to canonical methods. Managerial implications: Our method shows how to use an extreme spatial model for predicting extreme events and managing extreme risks in complex production systems such as aircraft. This can help achieve better quality management and operation safety in aircraft production systems and beyond.

We introduce two federated learning frameworks for the classical SPDnet model operating on symmetric positive definite (SPD) matrices with Stiefel-constrained parameters. Unlike standard Euclidean averaging, which violates orthogonality, our approach preserves geometric structure through two efficient aggregation strategies: ProjAvg, projecting arithmetic means onto the Stiefel manifold, and RLAvg, approximating tangent-space averaging via retractions and liftings. Both methods are computationally efficient, independent of the optimizer, and enable scalable federated learning for signal processing applications whose features are SPD matrices. Simulations on EEG motor imagery benchmarks show that FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation, while using fewer parameters per communication round.

The Pareto distribution plays a crucial role in various disciplines, necessitating robust goodness-of-fit tests for its validation. This article introduces a novel tests based on Stein's characterization and the Laplace transform, offering a fresh perspective on model assessment. We establish the asymptotic properties of the proposed test and evaluate its empirical performance against existing methods in terms of size and power. Our findings demonstrate that the new test often outperforms or performs comparably to established tests. In addition, real data applications illustrate its practical utility.

We investigate barycenters of Gaussian process laws in adapted Wasserstein space. The adapted Wasserstein distance refines classical optimal transport by enforcing compatibility of transport plans with the temporal flow of information, and is therefore well suited for stochastic systems with filtration constraints, as common in stochastic control, mathematical finance and sequential decision problems. Within this framework, we consider weighted Fréchet means of Gaussian process laws and prove that the associated barycenter problem admits Gaussian solutions. We derive a characterization of adapted Wasserstein barycenters in terms of the means and covariance operators of the underlying processes, and we analyze their existence, uniqueness, and regularity properties under natural assumptions. The Gaussian setting reveals a tractable and structurally rich subclass of adapted transport problems, bridging adapted optimal transport and Bures--Wasserstein geometry. Our results identify adapted Wasserstein barycenters as natural representatives of collections of Gaussian models and suggest new applications in stochastic optimization, robust finance, and sequential statistics.

We introduce Robust Bayesian Sequential Borrowing (RBSB), a framework for extrapolating evidence across adjacent subgroups in multi-population clinical programmes where studies are conducted in sequence and populations are ordered by clinical proximity. Conventional approaches weight all historical sources uniformly or exclude distant populations entirely, failing to reflect the natural gradient of similarity in such programmes. RBSB encodes the programme order through path-dependent borrowing via robust mixture priors that combine an informative component with a unit-information component to guard against prior-data conflict. Posterior weights, derived in closed form from marginal likelihood ratios, provide transparent dynamic attenuation when heterogeneity arises between sequential populations. The framework supports prospective evaluation of Bayesian Type I error, power, and extends naturally to assurance at both the study and programme level. Simulation studies demonstrate superior false-positive control relative to full pooling, while preserving substantial efficiency gains over standalone analyses. A case study of the START trial illustrates the approach across adult, adolescent, and paediatric populations. RBSB offers a practical, regulator-aligned method for disciplined evidence borrowing that exploits temporal and biological proximity while preventing implausible extrapolation across distant populations.

The Super Learner (SL) is a widely used ensemble method that combines predictions from a library of learners based on their predictive performance. Interval predictions are of considerable practical interest because they allow uncertainty in predictions produced by an individual learner or an ensemble to be quantified. Several methods have been proposed for constructing interval predictions based on the SL, however, these approaches are typically justified using asymptotic arguments or rely on computationally intensive procedures such as the bootstrap. Conformal prediction (CP) is a machine learning framework for constructing prediction intervals with finite-sample and asymptotic coverage guarantees under mild conditions. We propose coupling CP with the SL through a natural construction that mirrors the original SL framework, using individual learner weights and combining learner-specific conformity scores via a weighted majority vote. We characterize the properties of the resulting SL-based prediction intervals for continuous outcomes. We cover settings under exchangeability, potential violations of exchangeability, and data-generating mechanisms exhibiting heteroscedasticity, sparsity, and other forms of distributional heterogeneity. A comprehensive simulation study shows that the conformalized SL achieves valid finite-sample coverage with competitive performance relative to the true data-generating mechanism. A central contribution of this work is an application to predicting creatinine levels using socio-demographic, biometric, and laboratory measurements. This example demonstrates the benefits of an ensemble with carefully selected learners designed to capture key aspects of complex regression functions, including non-linear effects, interactions, sparsity, heteroscedasticity, and robustness to outliers.R

One of the major limits of kernel ridge regression (KRR) is that storing and manipulating the kernel matrix K_n for n samples requires O(n^2) space, which rapidly becomes unfeasible for large n. Nystrom approximations reduce the space complexity to O(nm) by sampling m columns from K_n. Uniform sampling preserves KRR accuracy (up to epsilon) only when m is proportional to the maximum degree of freedom of K_n, which may require O(n) columns for datasets with high coherence. Sampling columns according to their ridge leverage scores (RLS) gives accurate Nystrom approximations with m proportional to the effective dimension, but computing exact RLS also requires O(n^2) space. (Calandriello et al. 2016) propose INK-Estimate, an algorithm that processes the dataset incrementally and updates RLS, effective dimension, and Nystrom approximations on-the-fly. Its space complexity scales with the effective dimension but introduces a dependency on the largest eigenvalue of K_n, which in the worst case is O(n). In this paper we introduce SQUEAK, a new algorithm that builds on INK-Estimate but uses unnormalized RLS. As a consequence, the algorithm is simpler, does not need to estimate the effective dimension for normalization, and achieves a space complexity that is only a constant factor worse than exact RLS sampling.

Rejection sampling is a technique for sampling from difficult distributions. However, its use is limited due to a high rejection rate. Common adaptive rejection sampling methods either work only for very specific distributions or without performance guarantees. In this paper, we present pliable rejection sampling (PRS), a new approach to rejection sampling, where we learn the sampling proposal using a kernel estimator. Since our method builds on rejection sampling, the samples obtained are with high probability i.i.d. and distributed according to f. Moreover, PRS comes with a guarantee on the number of accepted samples.

We introduce and analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem in Euclidean space. For absolutely continuous source measures, this map is uniquely defined as the gradient of a convex function, a result known as Brenier's theorem. Without absolute continuity, the problem is relaxed, maps are replaced by coupling measures, and optimal couplings are supported on the subdifferential of a convex function, a Brenier potential. This characterization is the basis for our statistical estimator of Monge transport maps for measures known only through finite samples. The resulting Brenier potential has a simple closed-form expression based on the dual solution of the discrete sampled problem. In particular, our methodology does not rely on smoothness or continuity of the Monge transport map and requires no computation beyond primal-dual solutions of the discrete finite-dimensional problem. We exhibit convergence rates for this estimator based on a new error bound for the quadratic optimal transport problem. In the semi-discrete setting, where the target measure is finitely supported, our estimator enjoys sharper convergence rates. Finally, using similar proof techniques, we provide a novel convergence rate for empirical couplings.

Detecting copy number alterations (CNAs) from next-generation sequencing data remains challenging, particularly for short segments under noisy conditions. Existing segmentation methods often suffer from high false positive rates or fail to reliably detect short aberrations, especially in low-coverage data. In this study, we propose a modified tail-greedy unbalanced Haar (TGUHm) method that introduces a dual-thresholding strategy to improve segmentation accuracy. The proposed approach effectively suppresses spurious spikes while preserving sensitivity to both short and long CNA segments. Extensive simulation studies under Gaussian and heavy-tailed noise demonstrate that TGUHm consistently achieves higher true positive rates and lower false positive rates compared to state-of-the-art methods, including CBS, HaarSeg, and FDRSeg. In particular, the proposed method improves detection accuracy for short segments while maintaining competitive overall performance. Application to real cancer genomic data further confirms the practical utility of the method, revealing biologically meaningful CNAs associated with known cancer-related genes. These results suggest that TGUHm provides a robust and effective framework for CNA detection in challenging sequencing settings.

Classical ReLU-based Input Convex Neural Networks (ICNNs) are equivalent to the optimal value functions of Linear Programming (LP). This intrinsic structural equivalence restricts their representational capacity to piecewise-linear polyhedral functions. To overcome this representational bottleneck, we propose the SOC-ICNN, an architecture that generalizes the underlying optimization class from LP to Second-Order Cone Programming (SOCP). By explicitly injecting positive semi-definite curvature and Euclidean norm-based conic primitives, our formulation introduces native smooth curvature into the representation while preserving a rigorous optimization-theoretic interpretation. We formally prove that SOC-ICNNs strictly expand the representational space of ReLU-ICNNs without increasing the asymptotic order of forward-pass complexity. Extensive experiments demonstrate that SOC-ICNN substantially improves function approximation, while delivering competitive downstream decision quality. The code is available at https://github.com/Kanyooo/SOC-ICNN.

A nonparametric model using a sequence of Bernstein polynomials is constructed to approximate arbitrary isotropic covariance functions valid in $\mathbb{R}^\infty$ and related approximation properties are investigated using the popular $L_{\infty}$ norm and $L_2$ norms. A computationally efficient sieve maximum likelihood (sML) estimation is then developed to nonparametrically estimate the unknown isotropic covaraince function valid in $\mathbb{R}^\infty$. Consistency of the proposed sieve ML estimator is established under increasing domain regime. The proposed methodology is compared numerically with couple of existing nonparametric as well as with commonly used parametric methods. Numerical results based on simulated data show that our approach outperforms the parametric methods in reducing bias due to model misspecification and also the nonparametric methods in terms of having significantly lower values of expected $L_{\infty}$ and $L_2$ norms. Application to precipitation data is illustrated to showcase a real case study. Additional technical details and numerical illustrations are also made available.

Enhancing seismic fragility and risk assessment of nuclear power plants relies on accurate prediction of reactor building responses to seismic hazards, which can be further improved through dynamic analysis of high-fidelity finite element (FE) models. However, FE models often exhibit non-negligible discrepancies from actual structures due to various sources of uncertainty, necessitating FE model updating with rigorous quantification of associated uncertainties. This paper presents a GPU-accelerated latent space--based Bayesian framework for FE model updating of building structures. In the proposed framework, high-dimensional structural response data (e.g., time histories or frequency response functions) are projected into a low-dimensional latent space using a multimodal variational autoencoder (MVAE), thereby enabling efficient and tractable likelihood evaluation without explicit modeling in the original observation space. Once trained, the surrogate enables amortized inference, allowing posterior sampling to be performed without additional simulator evaluations. We specifically employ a sequential Monte Carlo (SMC) sampler, whose population-based formulation allows parallel evaluation of the approximate likelihood on GPUs, resulting in computational efficiency and robustness against multimodal and complex posterior distributions. The proposed framework is validated through both numerical benchmarking and experimental data from a shaking table test of a reinforced concrete building structure. The results demonstrate that the method accurately estimates structural parameters with well-quantified uncertainties, while achieving fast and efficient inference through GPU-based parallelization, and enabling robust inference even in the presence of sparse observations that induce multimodal and highly complex posterior distributions.

This paper studies expected performance and practical feasibility of the most commonly used classes of source-level likelihood-ratio (LR) systems when applied to a trace-reference comparison problem. The paper compares performance of these classes of LR systems (used to update prior odds) to each other and to the use of prior odds only, using strictly proper scoring rules as performance measures. It also explores practical feasibility of the classes of LR systems. The present analysis allows for a ranking of these classes of LR systems: from specific-source feature-based to common-source anchored or non-anchored score-based. A trade-off between performance and practical feasibility is observed, meaning that the best performing class of LR systems is the hardest to realise in practice, while the least performing class is the easiest to realise in practice. The other classes of LR systems are in between the two extremes. The one positive exception is a common-source feature-based LR system, with good performance and relatively low experimental demands. The paper also argues against the claim that some classes of LR systems should not be used, by showing that all systems have merit (when updating prior odds) over just using the prior odds (i.e. not using the LR system).

Most clinical prediction studies are developed from retrospective cohorts and reported as if all patient information were observed at once. In practice, clinicians face a more consequential question: \emph{when is there already enough information to stop testing and act?} A later stage can produce a better-looking model and still fail to justify the added delay, burden, or invasiveness of further workup. We formulate sequential clinical prediction as an \emph{optimal-stopping} problem under staged information, and illustrate the framework across four retrospective clinical datasets. The preferred stopping stage differed substantially by setting: sometimes fuller information justified waiting, whereas in other cases early or intermediate action was preferable. The key object is the patient-specific conditional risk trajectory: forward martingale structure represents coherent risk updating across stages, while reverse-martingale ideas describe information loss when a richer predictor is replaced by a simpler score. The results demonstrate that the best-performing model is not always the best stage for clinical decision-making.

Background: Minute-level accelerometer data capture rich diurnal physical activity (PA) patterns, but conventional summary metrics obscures clinically meaningful changes accumulated across a day. Building on Riemannian framework, we integrate multivariate functional principal component analysis (MFPCA) to identify main modes of PA change in older women and examine associations with physical function (PF). Method: A subset participant from OPACH as baseline and two WHISH follow-ups (W1, W2), yielded 3 accelerometer measurements; each participant's diurnal PA at each visit was represented as a smooth curve. Change between consecutive visits (defined as periods: baseline-W1, W1-W2) was modeled as a Riemannian deformation (RD) jointly capturing changes in PA timing and magnitude. Deformations were parameterized by initial momenta and summarized using MFPCA; participant-level changes were characterized by principal component (PC) scores and deformation energy (DE), a metric of overall pattern change. Associations with PF were assessed using linear mixed models. Results: Mean deformation in both periods showed overall downward shifts in PA magnitude with temporal redistribution between 10am and 7pm. Top 15 PCs explained >= 90% of variability in both periods; PC1 represented a pattern of PA increase/decrease throughout the day, explaining 22.4% (baseline-W1) and 20.8% (W1-W2). Among complete data (N=1157), an increase in PA in the mode of PC1 was positively associated with PF (p <0.0001). The interaction between DE and period was significantly associated with PF (p=0.003). Conclusions: Modeling longitudinal PA change as RDs and summarizing variability via MFPCA produced clinically interpretable phenotypes of diurnal PA change beyond standard metrics. The leading deformation mode was significantly associated with PF, and DE showed a stronger association with PF in the later period.

We propose a unified probabilistic framework for sparse count tensors with excess zeros, motivated by single-cell Hi-C data. The observed data are naturally represented as a three-way tensor indexed by genomic loci pairs and cells, exhibiting pronounced sparsity, zero inflation, and cell-to-cell heterogeneity. We introduce a zero-inflated Poisson tensor model that integrates low-rank CP structure, cluster-specific latent embeddings, and smoothness along ordered genomic loci, thereby jointly capturing multiway dependence, heterogeneity, and structured variation. We develop a Bayes-optimal procedure for distinguishing structural from technical zeros, enabling principled inference and uncertainty quantification. We establish identifiability of the model parameters and derive consistency rates for the proposed estimators in a high-dimensional regime. Simulation studies and analyses of single-cell Hi-C data demonstrate improved performance in zero detection, latent structure recovery, and downstream tasks such as clustering and 3D chromatin organization inference. The proposed framework provides a flexible approach for multiway count data with excess zeros and structured dependencies, and suggests several directions for future work, including mixture-based modeling of cell populations and scalable computation for large-scale applications.

Interaction effects are ubiquitous in applied statistical modelling, yet their meaningful interpretation remains challenging. The classic Johnson-Neyman (JN) technique (Johnson and Neyman 1936) addresses this challenge for two-way interactions by identifying the regions of a moderator's range over which a focal effect is and is not statistically significant. The int3ract package for R implements the JN technique and its three-way extension (the Johnson-Neyman-Krause, or JNK, technique) for both frequentist and Bayesian models. The function JNK_freq() auto-detects models fitted via lm()/glm(), RSiena's siena(), or lme4's lmer()/glmer(), but can also be applied to multiplicative interactions from (virtually) any model family by supplying a coefficient vector and covariance matrix directly. For Bayesian Stochastic Actor-Oriented Models (SAOMs) estimated with multiSiena, or any model producing posterior draws, the function JNK_bayes() produces conditional posterior distributions. For two-way interactions, classic shaded confidence-band plots are created that visually demarcate significant and non-significant regions along the moderator range; three-way interactions yield colour-gradient heatmaps with optional crosshatch overlays for non-significant regions. The package is designed to encourage richer, region-specific reporting of interaction effects in place of the conventional single-slope spotlight approach. The package is currently available on Github 'RWKrause/int3ract'.

In many longitudinal studies, a large number of variables are measured repeatedly over time, with substantial missing data. Existing methods, such as probabilistic principal component analysis (PPCA), are ill-equipped to handle such incomplete, high-dimensional longitudinal data, as they fail to account for the nested sources of variation and temporal dependency inherent in repeated measures. We introduce hierarchical probabilistic principal component analysis (HPPCA), a two-level probabilistic factor model that explicitly separates between-subject variance from time-varying within-subject dynamics. The within-subject latent factors are modeled by a Gaussian process. We develop an EM algorithm to handle missing data and flexible covariance kernels, accelerated by computationally efficient initializers. Simulation studies demonstrated that HPPCA robustly recovers model parameters subspaces and substantially outperforms both standard PPCA and multivariate functional PCA in imputation accuracy, even under heavy missingness and model misspecification. An application to the long COVID symptoms in the Researching COVID to Enhance Recovery adult cohort revealed that HPPCA effectively captured the data's hierarchical structure and its learned features significantly improved the prediction of clinical outcomes and the recovery of masked clinical records compared to exisiting methods.

Experimental designs that are minimax in the presence of model misspecifications have been constructed so as to minimize the maximum, over classes of alternate response models, of the integrated mean squared error of the predicted values. The theory to date has focussed almost exclusively on Least Squares estimates. Here we extend this theory to designs tailored for M-estimation of parameters, thus obtaining additional robustness against outlying responses. We show that, subject to a minor change in a tuning constant, designs optimal for Least Squares remain so asymptotically for M-estimation. We argue that even this minor change should be ignored, and the tuning constant chosen in an ad hoc but sensible manner which does not depend on which M-estimate is being employed. Our designs and estimates, derived under an assumption of i.i.d. errors, are also shown to be robust, in a minimax sense, against broad classes of correlation structures.

Network meta-analysis (NMA) combines direct and indirect comparisons across a connected treatment network to estimate relative treatment effects. However, there is a lack of exact contribution decompositions that reproduce NMA estimates, particularly in the presence of multi-arm trials that induce within-study correlations. We address this reproducibility gap by developing a contrast-space projection formulation of NMA. Working in the space of all estimable pairwise treatment contrasts, we express the NMA estimator as an explicit linear mapping of the observed contrasts onto the consistency-constrained contrast space induced by orthogonal projection. Building on this representation, we introduce a rigorous study-based definition of direct and indirect evidence through a canonical within-study reduction that removes algebraic redundancy and yields a unique, invariant decomposition. This leads to exact covariance-aware decompositions of the NMA estimator into study-level direct and indirect contributions, with indirect evidence further resolved into path-level components. The resulting weights are directly analogous to inverse-variance weights in pairwise meta-analysis and enable, to our knowledge, the first forest-plot representation that exactly reconstructs the NMA estimator. The framework also yields projection-based diagnostic and graphical tools, including forest plots, tension plots, and path-based visualizations. Applications to empirical datasets demonstrate how the proposed approach provides a reproducible and interpretable framework for understanding evidence contributions in network meta-analysis, supporting transparent interpretation and reporting.

This paper investigates the off-policy evaluation (OPE) problem from a distributional perspective. Rather than focusing solely on the expectation of the total return, as in most existing OPE methods, we aim to estimate the entire return distribution. To this end, we introduce a quantile-based approach for OPE using deep quantile process regression, presenting a novel algorithm called Deep Quantile Process regression-based Off-Policy Evaluation (DQPOPE). We provide new theoretical insights into the deep quantile process regression technique, extending existing approaches that estimate discrete quantiles to estimate a continuous quantile function. A key contribution of our work is the rigorous sample complexity analysis for distributional OPE with deep neural networks, bridging theoretical analysis with practical algorithmic implementations. We show that DQPOPE achieves statistical advantages by estimating the full return distribution using the same sample size required to estimate a single policy value using conventional methods. Empirical studies further show that DQPOPE provides significantly more precise and robust policy value estimates than standard methods, thereby enhancing the practical applicability and effectiveness of distributional reinforcement learning approaches.

Digital marketplaces processing billions of dollars annually represent critical infrastructure in sociotechnical ecosystems, yet their performance optimization lacks principled measurement frameworks that can inform algorithmic governance decisions regarding market efficiency and fairness from complex market data. By looking at orderbook data from double auction markets alone, because bids and asks do not represent true maximum willingnesses to buy and true minimum willingnesses to sell, there is little an economist can say about the market's actual performance in terms of allocative efficiency. We turn to experimental data to address this issue, `inverting' the standard induced value approach of double auction experiments. Our aim is to predict key market features relevant to market efficiency, particularly allocative efficiency, using orderbook data only -- specifically bids, asks and price realizations, but not the induced reservation values -- as early as possible. Since there is no established model of strategically optimal behavior in these markets, and because orderbook data is highly unstructured, non-stationary and non-linear, we propose quantile-based normalization techniques that help us build general predictive models. We develop and train several models, including linear regressions and gradient boosting trees, leveraging quantile-based input from the underlying supply-demand model. Our models can predict allocative efficiency with reasonable accuracy from the earliest bids and asks, and these predictions improve with additional realized price data. The performance of the prediction techniques varies by target and market type. Our framework holds significant potential for application to real-world market data, offering valuable insights into market efficiency and performance, even prior to any trade realizations.

Continual learning (CL) is concerned with learning multiple tasks sequentially without forgetting previously learned tasks. Despite substantial empirical advances over recent years, the theoretical development of CL remains in its infancy. At the heart of developing CL theory lies the challenge that the data distribution varies across tasks, and we argue that properly addressing this challenge requires understanding this variation--dependency among tasks. To explicitly model task dependency, we consider nonlinear regression tasks and propose the assumption that these tasks are dependent in such a way that the data of the current task is a nonlinear transformation of previous data. With this model and under natural assumptions, we prove statistical recovery guarantees (more specifically, bounds on estimation errors) for several CL paradigms in practical use, including experience replay with data-independent regularization and data-independent weights that balance the losses of tasks, replay with data-dependent weights, and continual learning with data-dependent regularization (e.g., knowledge distillation). To the best of our knowledge, our bounds are informative in cases where prior work gives vacuous bounds.

Bayesian filtering and smoothing for high-dimensional nonlinear dynamical systems are fundamental yet challenging problems in many areas of science and engineering. In this work, we propose FLUID, a flow-based unified amortized inference framework for filtering and smoothing dynamics. The core idea is to encode each observation history into a fixed-dimensional summary statistic and use this shared representation to learn both a forward flow for the filtering distribution and a backward flow for the backward transition kernel. Specifically, a recurrent encoder maps each observation history to a fixed-dimensional summary statistic whose dimension does not depend on the length of the time series. Conditioned on this shared summary statistic, the forward flow approximates the filtering distribution, while the backward flow approximates the backward transition kernel. The smoothing distribution over an entire trajectory is then recovered by combining the terminal filtering distribution with the learned backward flow through the standard backward recursion. By learning the underlying temporal evolution structure, FLUID also supports extrapolation beyond the training horizon. Moreover, by coupling the two flows through shared summary statistics, FLUID induces an implicit regularization across latent state trajectories and improves trajectory-level smoothing. In addition, we develop a flow-based particle filtering variant that provides an alternative filtering procedure and enables ESS-based diagnostics when explicit model factors are available. Numerical experiments demonstrate that FLUID provides accurate approximations of both filtering distributions and smoothing paths.

We propose Bayesian Univariate-Guided Sparse Regression (BUGS), a novel global-local shrinkage framework that incorporates marginal association information directly into the prior through a continuous modulation of shrinkage. Unlike existing approaches that treat predictors symmetrically or rely on post hoc screening, BUGS embeds univariate guidance within the nonlinear variance structure of a regularized horseshoe prior, inducing adaptive shrinkage that enhances signal-noise separation. We establish theoretical guarantees including prior concentration, posterior contraction, and guidance-induced shrinkage separation, while demonstrating robustness under uninformative guidance. To enable scalability in ultra-high dimensions, we develop BUGS-Active, an active-set MCMC approximation that restricts local updates to a data-adaptive subset A_n, reducing per-iteration complexity from O(p) to O(|A_n|) while preserving key theoretical properties such as sure screening and contraction. Empirically, the proposed framework achieves strong signal recovery together with substantially improved control of false discovery rates relative to existing methods. BUGS-Active scales to dimensions up to p = 1,000,000, and is applied to a DNA methylation study with n=1051 subjects and approximately 850,000 CpG sites, yielding accurate prediction and interpretable sparse selection. These results establish marginally guided shrinkage as a powerful and scalable paradigm for high-dimensional Bayesian inference.

We propose a novel framework for conducting causal inference based on counterfactual densities. While the current paradigm of causal inference is mostly focused on estimating average treatment effects (ATEs), which restricts the analysis to the first moment of the outcome variable, our density-based approach is able to detect causal effects based on general distributional characteristics. Following the Oaxaca-Blinder decomposition approach, we consider two types of counterfactual density effects that together explain observed discrepancies between the densities of the treated and control group. First, the distribution effect is the counterfactual effect of changing the conditional density of the control group to that of the treatment group, while keeping the covariates fixed at the treatment group distribution. Second, the covariate effect represents the effect of a hypothetical change in the covariate distribution. Both effects have a causal interpretation under the classical unconfoundedness and overlap assumptions. Methodologically, our approach is based on analyzing the conditional densities as elements of a Bayes Hilbert space, which preserves the non-negativity and integration-to-one constraints. We specify a flexible functional additive regression model estimating the conditional densities. We apply our method to analyze the German East--West income gap, i.e., the observed differences in wages between East Germans and West Germans. While most of the existing studies focus on the average differences and neglect other distributional characteristics, our density-based approach is suited to detect all nuances of the counterfactual distributions, including differences in probability masses at zero.

Sparse autoencoders can localize where concepts live in language models, but not how they interact during multi-step reasoning. We propose Causal Concept Graphs (CCG): a directed acyclic graph over sparse, interpretable latent features, where edges capture learned causal dependencies between concepts. We combine task-conditioned sparse autoencoders for concept discovery with DAGMA-style differentiable structure learning for graph recovery and introduce the Causal Fidelity Score (CFS) to evaluate whether graph-guided interventions induce larger downstream effects than random ones. On ARC-Challenge, StrategyQA, and LogiQA with GPT-2 Medium, across five seeds ($n{=}15$ paired runs), CCG achieves $\CFS=5.654\pm0.625$, outperforming ROME-style tracing ($3.382\pm0.233$), SAE-only ranking ($2.479\pm0.196$), and a random baseline ($1.032\pm0.034$), with $p<0.0001$ after Bonferroni correction. Learned graphs are sparse (5-6\% edge density), domain-specific, and stable across seeds.

We introduce Variational Joint Embedding (VJE), a reconstruction-free latent-variable framework for non-contrastive self-supervised learning in representation space. VJE maximizes a symmetric conditional evidence lower bound (ELBO) on paired encoder embeddings by defining a conditional likelihood directly on target representations, rather than optimizing a pointwise compatibility objective. The likelihood is instantiated as a heavy-tailed Student--\(t\) distribution on a polar representation of the target embedding, where a directional--radial decomposition separates angular agreement from magnitude consistency and mitigates norm-induced pathologies. The directional factor operates on the unit sphere, yielding a valid variational bound for the associated spherical subdensity model. An amortized inference network parameterizes a diagonal Gaussian posterior whose feature-wise variances are shared with the directional likelihood, yielding anisotropic uncertainty without auxiliary projection heads. Across ImageNet-1K, CIFAR-10/100, and STL-10, VJE is competitive with standard non-contrastive baselines under linear and \(k\)-NN evaluation, while providing probabilistic semantics directly in representation space for downstream uncertainty-aware applications. We validate these semantics through out-of-distribution detection, where representation-space likelihoods yield strong empirical performance. These results position the framework as a principled variational formulation of non-contrastive learning, in which structured feature-wise uncertainty is represented directly in the learned embedding space.

Unsupervised anomaly detection is a challenging problem due to the diversity of data distributions and the lack of labels. Ensemble methods are often adopted to mitigate these challenges by combining multiple detectors, which can reduce individual biases and increase robustness. Yet building an ensemble that is genuinely complementary remains challenging, since many detectors rely on similar decision cues and end up producing redundant anomaly scores. As a result, the potential of ensemble learning is often limited by the difficulty of identifying models that truly capture different types of irregularities. To address this, we propose a methodology for characterizing anomaly detectors through their decision mechanisms. Using SHapley Additive exPlanations, we quantify how each model attributes importance to input features, and we use these attribution profiles to measure similarity between detectors. We show that detectors with similar explanations tend to produce correlated anomaly scores and identify largely overlapping anomalies. Conversely, explanation divergence reliably indicates complementary detection behavior. Our results demonstrate that explanation-driven metrics offer a different criterion than raw outputs for selecting models in an ensemble. However, we also demonstrate that diversity alone is insufficient; high individual model performance remains a prerequisite for effective ensembles. By explicitly targeting explanation diversity while maintaining model quality, we are able to construct ensembles that are more diverse, more complementary, and ultimately more effective for unsupervised anomaly detection.

Ambitious decarbonisation targets are rapidly increasing the commission of new offshore wind farms. For these newly commissioned plants to run, accurate power forecasts are needed from the onset. These allow grid stability, good reserve management and efficient energy trading. Despite machine learning models having strong performances, they tend to require large volumes of site-specific data that new farms do not yet have. To overcome this data scarcity, we propose a novel transfer learning framework that clusters power output according to covariate meteorological features. Rather than training a single, general-purpose model, we thus forecast with an ensemble of expert models, each trained on a cluster. As these pre-trained models each specialise in a distinct weather pattern, they adapt efficiently to new sites and capture transferable, climate-dependent dynamics. Our contributions are two-fold - we propose this novel framework and comprehensively evaluate it on eight offshore wind farms, achieving accurate cross-domain forecasting with under five months of site-specific data. Our experiments achieve a MAE of 3.52\%, providing empirical verification that reliable forecasts do not require a full annual cycle. Beyond power forecasting, this climate-aware transfer learning method opens new opportunities for offshore wind applications such as early-stage wind resource assessment, where reducing data requirements can significantly accelerate project development whilst effectively mitigating its inherent risks.

As large language models (LLMs) transition from chat interfaces to integral components of stochastic pipelines and systems approaching general intelligence, the ability to faithfully sample from specified probability distributions has become a functional requirement rather than a theoretical curiosity. We present the first large-scale, statistically powered audit of native probabilistic sampling in frontier LLMs, benchmarking 11 models across 15 distributions. To disentangle failure modes, we employ a dual-protocol design: Batch Generation, where a model produces $N{=}1000$ samples within one response, and Independent Requests, comprising $N{=}1000$ stateless calls. We observe a sharp protocol asymmetry: batch generation achieves only modest statistical validity, with a 7% median pass rate, while independent requests collapse almost entirely, with 10 of 11 models passing none of the distributions. Beyond this asymmetry, we reveal that sampling fidelity degrades monotonically with distributional complexity and aggravates as the sampling horizon $N$ increases. Finally, we demonstrate how the propagation of these failures into downstream real-world application tasks introduces systematic biases: models fail to enforce uniform answer-position constraints in Multiple Choice Question generation and systematically violate demographic targets in attribute-constrained text-to-image prompt synthesis. These findings indicate that current LLMs lack a functional internal sampler, necessitating external tools for applications requiring statistical guarantees.

The reconstruction of time-dependent Robin coefficients is a challenging inverse heat transfer problem due to its inherent ill-posedness. This paper introduces a hierarchical Bayesian approach integrated with a persistent homology (PH) prior for robust coefficient estimation. By quantifying the birth and death of topological features, the PH-based prior provides a global structural constraint that transcends local derivative based penalties. Numerical experiments show that this topological perspective allows for the preservation of complex temporal profiles without the typical staircase distortions of total variation (TV) priors or the excessive blurring of Gaussian models. A key feature of our framework is the hierarchical implementation, which yields an automated, data-driven selection of hyperparameters. The results demonstrate that while PH-based inference yields competitive accuracy compared to TV regularization, it offers superior performance in preserving the multiscale characteristics of the Robin coefficient, providing a robust alternative for convective heat transfer diagnostics

Bayesian model updating provides a rigorous probabilistic framework for calibrating finite element (FE) models with quantified uncertainties, thereby enhancing damage assessment, response prediction, and performance evaluation of engineering structures. Recent advances in hierarchical Bayesian model updating (HBMU) enable robust parameter estimation under ill-posed/ill-conditioned settings and in the presence of inherent variability in structural parameters due to environmental and operational conditions. However, most HBMU approaches overlook multimodality in structural parameters that often arises when a structure experiences multiple damage states over its service life. This paper presents an HBMU framework that employs a Dirichlet process (DP) mixture prior on structural parameters (DP-HBMU). DP mixtures are nonparametric Bayesian models that perform clustering without pre-specifying the number of clusters, incorporating damage state classification into FE model updating. We formulate the DP-HBMU framework and devise a Metropolis-within-Gibbs sampler that draws samples from the posterior by embedding Metropolis updates for intractable conditionals due to the FE simulator. The applicability of DP-HBMU to damage localization is demonstrated through both numerical and experimental examples. We consider moment-resisting frame structures with beam-end fractures and apply the method to datasets spanning multiple damage states, from an intact state to moderate or severe damage state. The clusters inferred by DP-HBMU align closely with the assumed or observed damage states. The posterior distributions of stiffness parameters agree with ground truth values or observed fractures while exhibiting substantially reduced uncertainty relative to a non-hierarchical baseline. These results demonstrate the effectiveness of the proposed method in damage localization.

In a data matrix, we may distinguish between cases, each represented by a row vector for a statistical unit, and cells, which correspond to single entries of the data matrix. Recent developments in Robust Statistics have introduced the cellwise contamination paradigm, which assumes contamination on cells rather than on entire cases. This approach becomes particularly relevant as the number of variables increases. Indeed, discarding or downweighting entire cases because of a few anomalous cells in them, as done by traditional (casewise) robust methods, can result in substantial information loss, since the non-contaminated (or reliable) cells can still be highly informative. This philosophy can also be considered in fuzzy clustering, by assuming that reliable cells within a case may still provide useful information for determining fuzzy memberships. A robust fuzzy clustering proposal is thus introduced in this work, combining the advantages of dealing with outlying cells and simultaneously controlling the degree of fuzziness of unit assignments. The cluster-specific relationships among variables, detected by the fuzzy clustering approach, are also key to better identifying outlying cells and correct them. The strengths of the proposed methodology are illustrated through a simulation study and two real-world applications. The effects of the model's tuning parameters are explored, and some guidance for users on how to set them suitably is provided.

This paper introduces two quasi-metrics for performance assessment of multi-object tracking (MOT) algorithms. One quasi-metric is an extension of the generalised optimal subpattern assignment (GOSPA) metric and measures the discrepancy between sets of objects. The other quasi-metric is an extension of the trajectory GOSPA (T-GOSPA) metric and measures the discrepancy between sets of trajectories. Similar to the GOSPA-based metrics, these quasi-metrics include costs for localisation error for properly detected objects, the number of false objects and the number of missed objects. The T-GOSPA quasi-metric also includes a track switching cost. Differently from the GOSPA and T-GOSPA metrics, the proposed quasi-metrics have the flexibility of penalising missed and false objects with different costs, and the localisation costs are not required to be symmetric. We also explain how to obtain similarity score functions based on these quasi-metrics. The performance of several Bayesian MOT algorithms is assessed with the T-GOSPA quasi-metric via simulations.

It is crucial to learn the shared structures among functional predictors, as these structures characterize how predictor components exert common effects and, more generally, how predictors are homogeneously associated with the response. However, learning from multiple functional predictors is challenging because response-predictor dependencies may vary across representation dimensions and emerge at multiple resolutions, ranging from globally shared effects to predictor-specific effects. To address this issue, we propose a filtration-based shared structure learning framework for multiple functional predictors. The proposed framework organizes predictors through a hierarchical forest structure, in which shared and predictor-specific components are progressively identified from coarse to fine filtration layers. Building on this structure, we develop a filtration-based pursuit pipeline for shared structure discovery, together with a filtrated functional partial least squares method for shared component extraction and coefficient estimation under the learned shared structures. Simulation studies show that the proposed framework is able to recover the dominant coarse-to-fine organization of the underlying shared structures and yield improved prediction performance relative to competing methods. Applied to lower-limb angular kinematics, the proposed framework improves evaluation accuracy and reveals interpretable joint coordination patterns associated with aging. More broadly, it provides a new multiscale representation-learning perspective for complex data consisting of multiple multidimensional objects.

Probabilistic electricity price forecasting (PEPF) is vital for short-term electricity markets, yet the multivariate nature of day-ahead prices - spanning 24 consecutive hours - remains underexplored. At the same time, real-time decision-making requires methods that are both accurate and fast. We introduce an online algorithm for multivariate distributional regression models, allowing efficient modeling of the conditional means, variances, and dependence structures of electricity prices. The approach combines multivariate distributional regression with online coordinate descent and LASSO-type regularization (absolute shrinkage and selection operator), enabling scalable estimation in high-dimensional covariate spaces. Additionally, we propose a regularized estimation path over increasingly complex dependence structures, allowing for early stopping and avoiding overfitting. In a case study using historical data from the German day-ahead market, the proposed method yields interpretable and well-calibrated joint prediction intervals for the 24-dimensional price distribution and provides robust performance across a range of proper scoring rules. The results underscore the importance of modeling the dependence structure of electricity prices. Furthermore, we analyze the trade-off between predictive accuracy and computational costs for batch and online estimation and provide a high-performing open-source Python implementation in the ondil package.

We study online assortment optimization under stochastic choice when a decision maker simultaneously values cumulative revenue performance and the quality of post-hoc inference on revenue contrasts. We analyze a forced-exploration optimism-in-the-face-of-uncertainty (OFU) scheme that combines two regularized maximum-likelihood estimators: one based on all observations for sequential decision making, and one based only on exploration rounds for inference. Our general theory is developed under predictable score proxies and per-round action-dependent curvature domination. Under these conditions we establish a self-normalized concentration inequality, a likelihood-based ellipsoidal confidence-set theorem, and a regret bound for approximate optimistic actions that explicitly accounts for optimization error. For the multinomial logit (MNL) model we derive explicit score and curvature proxies and show that a balanced spaced singleton-exploration schedule yields realized coordinate coverage, implying regret $\Otilde(n_T + T/\sqrt{n_T})$ and revenue-contrast error $\Otilde(1/\sqrt{n_T})$ up to fixed problem-dependent factors. A hard two-assortment subclass yields a matching lower bound at the product level. Consequently, within the polynomial exploration family $n_T \asymp T^α$, the regret and inference rates become $\Otilde(T^{\max\{α,1-α/2\}})$ and $\Otilde(T^{-α/2})$, respectively; hence $α\in[2/3,1)$ is the rate-wise Pareto-undominated interval and $α=2/3$ is the unique balancing point that minimizes the regret exponent. Finally, for the Exponomial Choice and Nested Logit models we state verifiable sufficient conditions that would instantiate the general framework.

This paper considers filtering, parameter estimation, and testing for potentially dynamically misspecified state-space models. When dynamics are misspecified, filtered values of state variables often do not satisfy model restrictions, making them hard to interpret, and parameter estimates may fail to characterize the dynamics of filtered variables. To address this, a sequential optimal transportation approach is used to generate a model-consistent sample by mapping observations from a flexible reduced-form to the structural conditional distribution iteratively. Filtered series from the generated sample are model-consistent. Specializing to linear processes, a closed-form Optimal Transport Filtering algorithm is derived. Minimizing the discrepancy between generated and actual observations defines an Optimal Transport Estimator. Its large sample properties are derived. A specification test determines if the model can reproduce the sample path, or if the discrepancy is statistically significant. Empirical applications to DSGE models, affine term structure models, and trend-cycle decomposition illustrate the methodology and the results.

This study presents a scalable Bayesian estimation algorithm for sparse estimation in exploratory item factor analysis based on a classical Bayesian estimation method, namely Bayesian joint modal estimation (BJME). BJME estimates the model parameters and factor scores that maximize the complete-data joint posterior density. The algorithm's scalability is achieved through an alternating optimization scheme that iteratively updates model parameters and latent variables. Simulation studies show that the proposed algorithm has high computational efficiency and accuracy in variable selection over latent factors and the recovery of the model parameters. Moreover, we conducted a real data analysis using large-scale data from a psychological assessment that targeted the Big Five personality traits. This result indicates that the proposed algorithm achieves computationally efficient parameter estimation and extracts the interpretable factor loading structure.

This paper studies high-dimensional trend inference for piecewise smooth signals under nonstationary noise and asynchronous structural breaks by first detecting asynchronous changes without assuming stationarity and then further exploiting latent group structures to estimate trend functions. In the first step, we propose AJDN (Asynchronous Jump Detection under Nonstationary Noise), a multiscale framework for the identification and localization of jumps in high-dimensional time series. We show that AJDN consistently recovers the number of jumps with a prescribed asymptotic probability and achieves nearly optimal localization rates in the presence of asynchronicity and nonstationarity, both of which often violate the assumptions of existing high-dimensional change point methods and thereby deteriorate their performance. In the second step, we augment AJDN with a homogeneity pursuit step and obtain AJDN-H, which identifies latent groups of dimensions that share common jump structures and trend parameters given the detected jumps. This allows for efficient information pooling and improves the accuracy of trend estimation under both asynchronicity and nonstationarity. The robustness and finite-sample performance of the proposed methodology are examined by extensive simulation studies. An application to financial data demonstrates the practical utility of the AJDN-H framework in complex, high-dimensional settings.

Large-scale streaming data are common in modern machine learning applications and have led to the development of online learning algorithms. Many fields, such as supply chain management, weather and meteorology, energy markets, and finance, have pivoted toward probabilistic forecasting. This results in the need not only for accurate learning of the expected value but also for learning the conditional heteroskedasticity and conditional moments. Against this backdrop, we present a methodology for online estimation of regularized, linear distributional models. The proposed algorithm combines recent developments in online estimation of LASSO models with the well-known GAMLSS framework. We provide a case study on day-ahead electricity price forecasting, in which we show the competitive performance of the incremental estimation combined with strongly reduced computational effort. Our algorithms are implemented in a computationally efficient Python package ondil.

The maximum likelihood estimation is computationally demanding for large datasets, particularly when the likelihood function includes integrals. Subsampling can reduce the computational burden, but it often results in efficiency loss.This paper proposes a moment-assisted subsampling (MAS) method that can improve the estimation efficiency of existing subsampling-based maximum likelihood estimators.The motivation behind this approach stems from the fact that sample moments can be efficiently computed even if the sample size of the whole data set is huge.Through the generalized method of moments, the proposed method incorporates informative sample moments of the whole data. The MAS estimator can be computed rapidly and is asymptotically normal with a smaller asymptotic variance than the corresponding estimator without incorporating sample moments of the whole data. The asymptotic variance of the proposed estimator depends on the specific sample moments incorporated. We derive the optimal moment that minimizes the resulting asymptotic variance in terms of Loewner order. The proposed MAS estimator can achieve the same estimation efficiency as the whole data-based estimator when the optimal moment is incorporated. Numerical results demonstrate the promising performance of the proposed method in both estimation and computational efficiency compared with existing subsampling methods.

In recent years, a number of methods have been proposed to estimate the times at which a neuron spikes on the basis of calcium imaging data. However, quantifying the uncertainty associated with these estimated spikes remains an open problem. We consider a simple and well-studied model for calcium imaging data, which states that calcium decays exponentially in the absence of a spike, and instantaneously increases when a spike occurs. We wish to test the null hypothesis that the neuron did not spike -- i.e., that there was no increase in calcium -- at a particular timepoint at which a spike was estimated. In this setting, classical hypothesis tests lead to inflated Type I error, because the spike was estimated on the same data used for testing. To overcome this problem, we propose a selective inference approach. We describe an efficient algorithm to compute finite-sample p-values that control selective Type I error, and confidence intervals with correct selective coverage, for spikes estimated using a recent proposal from the literature. We apply our proposal in simulation and on calcium imaging data from the spikefinder challenge.

Many events and policies (treatments) occur at specific spatial locations, with researchers interested in their effects on nearby units. I approach the spatial treatment setting from an experimental perspective: What ideal experiment would we design to estimate the causal effects of spatial treatments? This perspective motivates a comparison between units near realized treatment locations and units near counterfactual (unrealized) candidate locations, which differs from current empirical practice. I derive design-based standard errors that are straightforward to compute. For observational data, I propose machine learning methods to find counterfactual candidate locations when observable characteristics, rather than potential outcomes, determine treatment probabilities. To accommodate methods for high-dimensional data in the theory, I extend a double machine learning result to the design-based framework with spatial correlations. I apply the proposed methods to study the causal effects of grocery stores on foot traffic to nearby businesses during COVID-19 shelter-in-place policies, finding a large positive effect at very short distances, with no effect at larger distances.