Papers

We consider the problem of inference for the states and parameters of a continuous-time multitype branching process from partially observed time series data. Exact inference for this class of models, typically using sequential Monte Carlo, can be computationally challenging when the populations that are being modelled grow exponentially or the time series is long. Instead, we derive a Gaussian approximation for the transition function of the process that leads to a Kalman filtering algorithm that runs in a time independent of the population sizes. We also develop a hybrid approach for when populations are smaller and the approximation is less applicable. We investigate the performance of our approximation and algorithms to both a simple and a complex epidemic model, finding good adherence to the true posterior distributions in both cases with large computational speed-ups in most cases. We also apply our method to a COVID-19 dataset with time dependent parameters where exact methods are intractable due to the population sizes involved.

The Jeffreys-Lindley paradox stands as the most profound divergence between frequentist and Bayesian approaches to hypothesis testing. Yet despite more than six decades of discussion, this paradox remains frequently misunderstood–even in the pages of leading statistical journals. In a 1993 paper published in Statistica Sinica, Robert characterized the Jeffreys-Lindley paradox as "the fact that a point null hypothesis will always be accepted when the variance of a conjugate prior goes to infinity." This characterization, however, describes a different phenomenon entirely-what we term Bartlett's Anomaly-rather than the Jeffreys-Lindley paradox as originally formulated. The paradox, as presented by Lindley (1957), concerns what happens as sample size increases without bound while holding the significance level fixed, not what happens as prior variance diverges. This distinction is not merely terminological: the two phenomena have different mathematical structures, different implications, and require different solutions. The present paper aims to clarify this confusion, demonstrating through Lindley's own equations that he was concerned exclusively with sample size asymptotics. We show that even Jeffreys himself underestimated the practical frequency of the paradox. Finally, we argue that the only genuine resolution lies in abandoning point null hypotheses in favor of interval nulls, a paradigm shift that eliminates the paradox and restores harmony between Bayesian and frequentist inference. Submitted to Statistica Sinica.

Evaluation is the foundation of empirical science, yet the evaluation of evaluation itself – so-called meta-evaluation – remains strikingly underdeveloped. While methods such as observational studies, design of experiments (DoE), and randomized controlled trials (RCTs) have shaped modern scientific practice, there has been little systematic inquiry into their comparative validity and utility across domains. Here we introduce a formal framework for meta-evaluation by defining the evaluation space, its structured representation, and a benchmark we call AxiaBench. AxiaBench enables the first large-scale, quantitative comparison of ten widely used evaluation methods across eight representative application domains. Our analysis reveals a fundamental limitation: no existing method simultaneously achieves accuracy and efficiency across diverse scenarios, with DoE and observational designs in particular showing significant deviations from real-world ground truth. We further evaluate a unified method of entire-space stratified sampling from previous evaluatology research, and the results report that it consistently outperforms prior approaches across all tested domains. These results establish meta-evaluation as a scientific object in its own right and provide both a conceptual foundation and a pragmatic tool set for advancing trustworthy evaluation in computational and experimental research.

Reinforcement learning in mobile health (mHealth) interventions requires balancing intervention efficacy with user burden, particularly when state measurements (for example, user surveys or feedback) are costly yet essential. The Act-Then-Measure (ATM) heuristic addresses this challenge by decoupling control and measurement actions within the Action-Contingent Noiselessly Observable Markov Decision Process (ACNO-MDP) framework. However, the standard ATM algorithm relies on a temporal-difference-inspired Q-learning method, which is prone to instability in sparse and noisy environments. In this work, we propose a Bayesian extension to ATM that replaces standard Q-learning with a Kalman filter-style Bayesian update, maintaining uncertainty-aware estimates of Q-values and enabling more stable and sample-efficient learning. We evaluate our method in both toy environments and clinically motivated testbeds. In small, tabular environments, Bayesian ATM achieves comparable or improved scalarized returns with substantially lower variance and more stable policy behavior. In contrast, in larger and more complex mHealth settings, both the standard and Bayesian ATM variants perform poorly, suggesting a mismatch between ATM's modeling assumptions and the structural challenges of real-world mHealth domains. These findings highlight the value of uncertainty-aware methods in low-data settings while underscoring the need for new RL algorithms that explicitly model causal structure, continuous states, and delayed feedback under observation cost constraints.

We study online statistical inference for the solutions of stochastic optimization problems with equality and inequality constraints. Such problems are prevalent in statistics and machine learning, encompassing constrained $M$-estimation, physics-informed models, safe reinforcement learning, and algorithmic fairness. We develop a stochastic sequential quadratic programming (SSQP) method to solve these problems, where the step direction is computed by sequentially performing a quadratic approximation of the objective and a linear approximation of the constraints. Despite having access to unbiased estimates of population gradients, a key challenge in constrained stochastic problems lies in dealing with the bias in the step direction. As such, we apply a momentum-style gradient moving-average technique within SSQP to debias the step. We show that our method achieves global almost-sure convergence and exhibits local asymptotic normality with an optimal primal-dual limiting covariance matrix in the sense of Hájek and Le Cam. In addition, we provide a plug-in covariance matrix estimator for practical inference. To our knowledge, the proposed SSQP method is the first fully online method that attains primal-dual asymptotic minimax optimality without relying on projection operators onto the constraint set, which are generally intractable for nonlinear problems. Through extensive experiments on benchmark nonlinear problems, as well as on constrained generalized linear models and portfolio allocation problems using both synthetic and real data, we demonstrate superior performance of our method, showing that the method and its asymptotic behavior not only solve constrained stochastic problems efficiently but also provide valid and practical online inference in real-world applications.

We present a Fourier-enhanced recurrent neural network (RNN) for downscaling electrical loads. The model combines (i) a recurrent backbone driven by low-resolution inputs, (ii) explicit Fourier seasonal embeddings fused in latent space, and (iii) a self-attention layer that captures dependencies among high-resolution components within each period. Across four PJM territories, the approach yields RMSE lower and flatter horizon-wise than classical Prophet baselines (with and without seasonality/LAA) and than RNN ablations without attention or Fourier features.

Kolmogorov–Arnold Networks (KANs) offer a promising path toward interpretable machine learning: their learnable activations can be studied individually while collectively fitting complex data accurately. In practice, however, trained activations often lack *symbolic fidelity*, learning pathological decompositions with no meaningful correspondence to interpretable forms. We propose Softly Symbolified Kolmogorov–Arnold Networks (S2KAN), which integrate symbolic primitives directly into training. Each activation draws from a dictionary of symbolic and dense terms, with learnable gates that sparsify the representation. Crucially, this sparsification is differentiable, enabling end-to-end optimization, and uses an optimization objective inspired by minimum description length. S2KAN can discover interpretable activation functions while retaining dense splines for flexible approximation. We introduce a quantitative measure of symbolic fidelity, assessing how closely learned activations can be approximated by short expressions from a specified symbolic language. Across real-world prediction tasks, dynamical-system forecasting, and synthetic symbolic benchmarks, S2KAN generally achieves higher symbolic fidelity and often far smaller models, while maintaining competitive predictive accuracy on real-world data and low prediction errors on most symbolic benchmark configurations. We also examine the models' learned responses on scientific data, including the contributions of their symbolic components. Together, these results show that incorporating symbolic structure during training can yield compact, accurate KAN models whose learned responses can be examined through their symbolic components.

This chapter opens with a review of classic tools for regression, a subset of machine learning that seeks to find relationships between variables. With the advent of scientific machine learning this field has moved from a purely data-driven (statistical) formalism to a constrained or “physics-informed” formalism, which integrates physical knowledge and methods from traditional computational engineering. In the first part, we introduce the general concepts and the statistical flavor of regression versus other forms of curve fitting. We then move to an overview of traditional methods from machine learning and their classification and ways to link these to traditional computational science. Finally, we close with a note on methods to combine machine learning and numerical methods for physics

This chapter provides three tutorial exercises on physics-constrained regression. These are implemented as toy problems that seek to mimic grand challenges in (1) the super-resolution and data assimilation of the velocity field in image velocimetry, (2) data-driven turbulence modeling, and (3) system identification and digital twinning for forecasting and control. The Python codes for all exercises are provided in the course repository.

As COVID-19 transitions into an endemic disease that remains constantly present in the population at a stable level, monitoring its prevalence without invasive measures becomes increasingly important. In this paper, we present a deep neural network estimator for the COVID-19 daily case count based on wastewater surveillance data and other confounding factors. This work builds upon the study by Jiang, Kolozsvary, and Li (2024), which connects the COVID-19 case counts with testing data collected early in the pandemic. Using the COVID-19 testing data and the wastewater surveillance data during the period when both data were highly reliable, one can train an artificial neural network that learns the nonlinear relation between the COVID-19 daily case count and the wastewater viral RNA concentration. From a machine learning perspective, the main challenge lies in addressing temporal feature reliability, as the training data has different reliability over different time periods.

The past decade has seen a massive rise in the popularity of AI systems, mainly owing to the developments in Gen AI, which has revolutionized numerous industries and applications. However, this progress comes at a considerable cost to the environment as training and deploying these models consume significant computational resources and energy and are responsible for large carbon footprints in the atmosphere. In this paper, we study the amount of carbon dioxide released by models across different domains over varying time periods. By examining parameters such as model size, repository activity (e.g., commits and repository age), task type, and organizational affiliation, we identify key factors influencing the environmental impact of AI development. Our findings reveal that model size and versioning frequency are strongly correlated with higher emissions, while domain-specific trends show that NLP models tend to have lower carbon footprints compared to audio-based systems. Organizational context also plays a significant role, with university-driven projects exhibiting the highest emissions, followed by non-profits and companies, while community-driven projects show a reduction in emissions. These results highlight the critical need for green AI practices, including the adoption of energy-efficient architectures, optimizing development workflows, and leveraging renewable energy sources. We also discuss a few practices that can lead to a more sustainable future with AI, and we end this paper with some future research directions that could be motivated by our work. This work not only provides actionable insights to mitigate the environmental impact of AI but also poses new research questions for the community to explore. By emphasizing the interplay between sustainability and innovation, our study aims to guide future efforts toward building a more ecologically responsible AI ecosystem.

This paper proposes a novel test method for high-dimensional mean testing regard for the temporal dependent data. Comparison to existing methods, we establish the asymptotic normality of the test statistic without relying on restrictive assumptions, such as Gaussian distribution or M-dependence. Importantly, our theoretical framework holds potential for extension to other high-dimensional problems involving temporal dependent data. Additionally, our method offers significantly reduced computational complexity, making it more practical for large-scale applications. Simulation studies further demonstrate the computational advantages and performance improvements of our test.

Given $n$ i.i.d. observations, we study the problem of estimating the spectrum of weighted Laplace operators of the form $\Delta_f=\Delta + \alpha \nabla \log f\cdot \nabla$, where $f$ is a positive probability density on a known compact $d$-dimensional manifold without boundary and $\alpha\in \mathbb{R}$ is a hyperparameter. These operators arise as continuum limits of graph Laplacian matrices and provide valuable geometric information on the underlying data distribution. We establish the exact minimax rates of estimation for this problem, by exhibiting two different rates of convergence for eigenfunctions and eigenvalues. When $f$ belongs to a Hölder-Zygmund class $\mathscr{C}^s$ of regularity $s\geqslant 2$, the eigenfunctions can be estimated with respect to the $\mathrm{L}^q$-norm ($q\geqslant 1$) via plug-in methods at the minimax rate $n^{-\frac{s+1}{2s+d}}$ for $d\geqslant 3$ (with different rates for $d\leqslant 2$). Moreover, eigenvalues can be estimated at the minimax rate $n^{-\frac{4s}{4s+d}}+n^{-\frac 12}$. In the regime $s>\frac d4$, we further show that asymptotically efficient estimators exist. We also present a general framework for estimating nonlinear functionals over Hölder-Zygmund spaces, with potential applications to a broad class of statistical problems.

This paper explains why internal and external validity cannot be simultaneously maximised. It introduces "evidential states" to represent the information available for causal inference and shows that routine study operations (restriction, conditioning, and intervention) transform these states in ways that do not commute. Because each operation removes or reorganises information differently, changing their order yields evidential states that support different causal claims. This non-commutativity creates a structural trade-off: the steps that secure precise causal identification also eliminate the heterogeneity required for generalisation. Small model, observational and experimental examples illustrate how familiar failures of transportability arise from this order dependence. The result is a concise structural account of why increasing causal precision necessarily narrows the world to which findings apply.

Time series data often contain initial transient periods before reaching a stable state, posing challenges in analysis and interpretation. In this paper, we propose a novel approach to detect and estimate the end of the initial transient in time series data. Our method leverages the reversal mean standard error (RMSE) as a metric for assessing the stability of the data. Additionally, we employ fractional filtering techniques to enhance the detection accuracy by filtering out noise and capturing essential features of the underlying dynamics. Combining with autocorrelation-corrected confidence intervals we provide a robust framework to automate transient detection and convergence assessment. The method ensures statistical rigor by accounting for autocorrelation effects, validated through simulations with varying time steps. Results demonstrate independence from numerical parameters (e.g., time step size, under-relaxation factors), offering a reliable tool for steady-state analysis. The framework is lightweight, generalizable, and mitigates inflated false positives in autocorrelated datasets.

In an earlier joint work, we studied a sequential Monte Carlo algorithm to sample from the Gibbs measure supported on torus with a non-convex energy function at a low temperature, where we proved that the time complexity of the algorithm is polynomial in the inverse temperature. However, the analysis in that torus setting relied crucially on compactness and does not directly extend to unbounded domains. This work introduces a new approach that resolves this issue and establishes a similar result for sampling from Gibbs measures supported on Rd. In particular, our main result shows that for double-well energy with equal well depths, the time complexity scales as seventh power of the inverse temperature, and quadratically in both the inverse allowed absolute error and probability error.

This study addresses the critical challenge of modeling and mapping urban air quality to ascertain pollutant concentrations in unmonitored locations. The advent of low-cost sensors, particularly those deployed in vehicular networks, presents novel datasets that hold the potential to enhance air quality modeling. This research conducts a comprehensive review of ten statistical models drawn from existing literature, using both fixed and mobile low-cost sensor data, alongside ancillary variables, within the urban confines of Nantes, France. Employing a methodology that includes cross-validation of data from low-cost sensors and validation on fixed air quality monitoring stations, this paper evaluates the models' performance in scenarios of temporal interpolation and prediction. Our findings reveal a pronounced bias in the model outputs when reliant on low-cost sensor data compared to the verification data obtained from fixed stations. Furthermore, machine learning models demonstrated superior performance in predictive scenarios, suggesting their enhanced suitability for forecasting tasks. The study conclusively indicates that reliance solely on data from low-cost mobile sensors compromises the reliability of air quality models, due to significant accuracy deficiencies. Consequently, we advocate for a directed focus towards the integration and calibration of low-cost sensor data with information from fixed monitoring stations. This approach, rather than an exclusive emphasis on the complexity of statistical modeling techniques, is pivotal for achieving the precision required for effective air quality management and policy-making.

With the advancement of technology and the arrival of miniaturized environmental sensors that offer greater performance, the idea of building mobile network sensing for air quality has quickly emerged to increase our knowledge of air pollution in urban environments. However, with these new techniques, the difficulty of building mathematical models capable of aggregating all these data sources in order to provide precise mapping of air quality arises. In this context, we explore the spatio-temporal geostatistics methods as a solution for such a problem and evaluate three different methods: Simple Kriging (SK) in residuals, Ordinary Kriging (OK), and Kriging with External Drift (KED). On average, geostatistical models showed 26.57% improvement in the Root Mean Squared Error (RMSE) compared to the standard Inverse Distance Weighting (IDW) technique in interpolating scenarios (27.94% for KED, 26.05% for OK, and 25.71% for SK). The results showed less significant scores in extrapolating scenarios (a 12.22% decrease in the RMSE for geostatisical models compared to IDW). We conclude that univariable geostatistics is suitable for interpolating this type of data but is less appropriate for an extrapolation of non-sampled places since it does not create any information.

The Hawkes process is a versatile stochastic model for point patterns that exhibit self-excitation, that is, the property that an event occurrence increases the rate of occurrence for some period of time in the future. We present a Bayesian nonparametric modeling approach for temporal marked Hawkes processes. Our focus is on point process modeling of earthquake occurrences, where the mark variable is given by earthquake magnitude. We develop a nonparametric prior model for the marked Hawkes process excitation function, using a representation with basis components for the time lag and the mark, and basis weights defined through a gamma process prior. We elaborate the model with a nonparametric prior for time-dependent background intensity functions, thus enabling a fully nonparametric approach to modeling the ground process intensity of marked Hawkes processes. The model construction balances computationally tractable inference with flexible forms for marked Hawkes process functionals, including mark-dependent offspring densities. The posterior simulation method provides full inference, without any approximations to the Hawkes process likelihood. In the context of the application, the modeling approach enables estimation of aftershock densities that vary with the magnitude of the main shock, thus significantly expanding the inferential scope of existing self-exciting point process models for earthquake occurrences. We investigate different aspects of the methodology through study of model properties, and with inference results based on synthetic marked point patterns. The practical utility of modeling magnitude-dependent aftershock dynamics is demonstrated with analysis of earthquakes that occurred in Japan from 1885 through 1980.

Bayesian hypothesis testing via Bayes factors offers a principled alternative to classical p-value methods in meta-analysis, particularly suited to its cumulative and sequential nature. Unlike commonly reported p-values for standard null hypothesis significance testing, Bayes factors allow for quantifying support both for and against the existence of an effect, facilitate ongoing evidence monitoring, and maintain coherent long-run behavior as additional studies are incorporated. Recent theoretical developments further show how Bayes factors can flexibly control Type I error rates through connections to e-value theory. Despite these advantages, their use remains limited in the meta-analytic literature. This paper provides a critical overview of their theoretical properties, methodological considerations, such as prior sensitivity, and practical advantages for evidence synthesis. Two illustrative applications are provided: one on statistical learning in individuals with language impairments, and another on seroma incidence following post-operative exercise in breast cancer patients. New tools supporting these methods are available in the open-source R package BFpack.

Causal inference, as a major research area in statistics and data science, plays a central role across diverse fields such as medicine, economics, education, and the social sciences. Design-based causal inference begins with randomized experiments and emphasizes conducting statistical inference by leveraging the known randomization mechanism, thereby enabling identification and estimation of causal effects under weak model dependence. Grounded in the seminal works of Fisher and Neyman, this paradigm has evolved to include various design strategies, such as stratified randomization and rerandomization, and analytical methods including Fisher randomization tests, Neyman-style asymptotic inference, and regression adjustment. In recent years, with the emergence of complex settings involving high-dimensional data, individual noncompliance, and network interference, design-based causal inference has witnessed remarkable theoretical and methodological advances. This paper provides a systematic review of recent progress in this field, focusing on covariate-balanced randomization designs, design-based statistical inference methods, and their extensions to high-dimensional, noncompliance, and network interference scenarios. It concludes with a comprehensive perspective on future directions for the theoretical development and practical applications of causal inference.

This paper explores Vecchia likelihood approximation for modeling physical phenomena sensed by mobile and fixed low-cost sensors in urban environments. A three-level hierarchical model is proposed to simultaneously accounts for the physical process of interest and measurement errors inherent in low-cost sensors. Several innovative configurations of Vecchia's approximation are investigated, including variations in ordering strategies, distance definitions, and sensor-specific conditioning. These configurations are evaluated for approximating the likelihood of a spatio-temporal Gaussian process, using simulated data based on real mobile sensor trajectories across Nantes, France. Our findings highlight the effectiveness of the min-max distance algorithm for ordering, reaffirming existing literature. Additionally, we demonstrate the utility of a random ordering approach that doesn't require prior definition of a spatio-temporal distance. These two ordering configurations achieved, on average, 102% better results in log Kullback-Leibler divergence compared with four other ordering schemes studied. Results are supplemented with Asymptotic Relative Efficiency analysis, offering practical recommendations for optimizing parameter estimation. The proposed model and preferred Vecchia configuration are applied to real-world air quality data collected using mobile and fixed low-cost sensors. This application underscores the model's practical value for pollution mapping and prediction in environmental monitoring. This study advances the use of Vecchia's approximation for addressing computational challenges of Gaussian models in large-scale spatio-temporal datasets from environmental monitoring with low-cost sensor networks.

Ranking methods or models based on their performance is of prime importance but is tricky because performance is fundamentally multidimensional. In the case of classification, precision and recall are scores with probabilistic interpretations that are both important to consider and complementary. The rankings induced by these two scores are often in partial contradiction. In practice, therefore, it is extremely useful to establish a compromise between the two views to obtain a single, global ranking. Over the last fifty years or so, it has been proposed to take a weighted harmonic mean, known as the F-score, F-measure, or $F_\beta$. Generally speaking, by averaging basic scores, we obtain a score that is intermediate in terms of values. However, there is no guarantee that these scores lead to meaningful rankings and no guarantee that the rankings are good tradeoffs between these base scores. Given the ubiquity of $F_\beta$ scores in the literature, some clarification is in order. Concretely: (1) We establish that $F_\beta$-induced rankings are meaningful and define a shortest path between precision- and recall-induced rankings. (2) We frame the problem of finding a tradeoff between two scores as an optimization problem expressed with Kendall rank correlations. We show that $F_1$ and its skew-insensitive version are far from being optimal in that regard. (3) We provide theoretical tools and a closed-form expression to find the optimal value for $\beta$ for any distribution or set of performances, and we illustrate their use on six case studies. Code is available at https://github.com/pierard/cvpr-2026-optimal-tradeoff-precision-recall.

We have developed a new signature-based spatial scan statistic for functional data (SigFSS). This scan statistic can be applied to both univariate and multivariate functional data. In a simulation study, SigFSS almost always performed better than the literature approaches and yielded more precise clusters in geographic terms. Lastly, we used SigFSS to search for spatial clusters of abnormally high or abnormally low mortality rates in mainland France.

Advances in tracking technologies for animal movement require new statistical tools to better exploit the increasing amount of data. Animal positions are usually calculated using the GPS or Argos satellite system and include potentially non-Gaussian and heavy-tailed measurement error patterns. Errors are usually handled through a Kalman filter algorithm, which can be sensitive to non-Gaussian error distributions. We introduce a latent movement model through an underdamped Langevin stochastic differential equation (SDE) that includes an additional drift term to ensure that the animal remains in a known spatial domain of interest. This can be applied to aquatic animals moving in water or terrestrial animals moving in a restricted zone delimited by fences or natural barriers. We demonstrate that the incorporation of these spatial constraints into the latent movement model can improve the accuracy of filtering for noisy observations of the positions. We implement an Extended Kalman Filter as well as a particle filter adapted to non-Gaussian error distributions. Our filters are based on solving the SDE through splitting schemes to approximate the latent dynamic. We illustrate the approach on a real Argos telemetry track of a bowhead whale in Foxe Basin, Canada.

We developed two new alternatives to signature-based, spatial autoregressive models. In a simulation study, we found that the new models performed at least as well as existing approaches but presented shorter computation times. We then used the new models to analyze the premature mortality rate and the mortality rate for people aged 65 and over.

Bayesian computational strategies for inference can be inefficient in approximating the posterior distribution in models that exhibit some form of periodicity. This is because the probability mass of the marginal posterior distribution of the parameter representing the period is usually highly concentrated in a very small region of the parameter space. Therefore, it is necessary to provide as much information as possible to the inference method through the parameter prior distribution. We intend to show that it is possible to construct a prior distribution from the data by fitting a Gaussian process (GP) with a periodic kernel. More specifically, we want to show that it is possible to approximate the marginal posterior distribution of the hyperparameter corresponding to the period in the kernel. Subsequently, this distribution can be used as a prior distribution for the inference method. We use an adaptive importance sampling method to approximate the posterior distribution of the hyperparameters of the GP. Then, we use the marginal posterior distribution of the hyperparameter related to the periodicity in order to construct a prior distribution for the period of the parametric model. This workflow is empirical Bayes, implemented as a modular (cut) transfer of a GP posterior for the period to the parametric model. We applied the proposed methodology to both synthetic and real data. We approximated the posterior distribution of the period of the GP kernel and then passed it forward as a posterior-as-prior with no feedback. Finally, we analyzed its impact on the marginal posterior distribution.

Wasserstein autoregression provides a robust framework for modeling serial dependence among probability distributions, with wide-ranging applications in economics, finance, and climate science. In this paper, we develop portmanteau-type diagnostic tests for assessing the adequacy of Wasserstein autoregressive models. By defining autocorrelation functions for model errors and residuals in the Wasserstein space, we construct two related tests: one analogous to the classical McLeod type test, and the other based on the sample-splitting approach of Davis and Fernandes(2025). We establish that, under mild regularity conditions, the corresponding test statistics converge in distribution to chi-square limits. Simulation studies and empirical applications demonstrate that the proposed tests effectively detect model mis-specification, offering a principled and reliable diagnostic tool for distributional time series analysis.

Selecting the best alternative from a finite set is the central objective of ranking and selection (R&S) and best arm identification (BAI). Traditional R&S or BAI approaches have predominantly relied on Gaussian or sub-Gaussian assumptions on the performance distributions of all alternatives, which limit their applicability to non-sub-Gaussian—especially heavy-tailed—problems. The need to move beyond sub-Gaussianity may become even more critical in large-scale problems, which tend to be especially sensitive to distributional specifications. In this paper, motivated by the widespread use of upper confidence bound (UCB) algorithms in sequential decision making, we investigate their performance in large-scale, non-sub-Gaussian R&S settings. We consider the simplest category of UCB algorithms, where the UCB value for each alternative is defined as the sample mean plus an exploration bonus that depends only on its own sample size. We abstract this into a meta-UCB algorithm and propose letting it select the alternative with the largest sample size as the best upon stopping. For this meta-UCB algorithm, we first derive a distribution-free lower bound on the probability of correct selection. Building on this bound, we first study the indifference-zone formulation and show that the meta-UCB algorithm—and therefore a broad class of UCB algorithms—achieves sample optimality as long as the variances are uniformly bounded. We then extend sample optimality to a non-indifference-zone configuration with polynomially shrinking mean gaps under the same variance assumption. These results demonstrate the applicability of UCB algorithms to large-scale R&S problems with non-sub-Gaussian distributions. Numerical experiments support our results and provide additional insights into the comparative behaviors of UCB algorithms within and beyond our meta-UCB framework.

We provide a survey of how techniques developed for the modelling of extremes naturally matter in insurance, and how they need to and can be adapted for the insurance applications. Topics covered include truncation, tempering, censoring and regression techniques. The discussed techniques are illustrated on concrete data sets.