We revisit size controllability results in Pötscher and Preinerstorfer (2025) concerning heteroskedasticity robust test statistics in regression models. For the special, but important, case of testing a single restriction (e.g., a zero restriction on a single coefficient), we povide a necessary and sufficient condition for size controllability, whereas the condition in Pötscher and Preinerstorfer (2025) is, in general, only sufficient (even in the case of testing a single restriction).

This paper develops and analyzes three families of estimators that continuously interpolate between classical quantiles and the sample mean. The construction begins with a smoothed version of the $L_1$ loss, indexed by a location parameter $z$ and a smoothing parameter $h \ge 0$, whose minimizer $\hat q(z,h)$ yields a unified $M$-estimation framework. Depending on how $(z, h)$ is specified, this framework generates three distinct classes of estimators: fixed-parameter smoothed quantile estimators, plug -- in estimators of fixed quantiles, and a new continuum of mean -- estimating procedures. For all three families we establish consistency and asymptotic normality via a uniform asymptotic equicontinuity argument. The limiting variances admit closed forms, allowing a transparent comparison of efficiency across families and smoothing levels. A geometric decomposition of the parameter space shows that, for fixed quantile level $τ$, admissible pairs $(z, h)$ lie on straight lines along which the estimator targets the same population quantile while its asymptotic variance evolves. The theoretical analysis reveals two efficiency regimes. Under light-tailed distributions (e.g., Gaussian), smoothing yields a monotone variance reduction. Under heavy-tailed distributions (e.g., Laplace), a finite smoothing parameter $h^{*}(τ) > 0$ strictly improves efficiency for quantile estimation. Numerical experiments -- based on simulated data and real financial returns -- validate these conclusions and show that, both asymptotically and in finite samples, the mean-estimating family does not improve upon the sample mean.

Recent advances in deep learning have highlighted the phenomenon of benign overfitting in overparameterized statistical models, sparking significant interest in understanding its foundations. Owing to its simplicity and practical relevance, the ordinary least squares (OLS) interpolator has become a key object of study for gaining theoretical insight into this phenomenon. While the properties of OLS are well understood in classical underparameterized settings, its behavior in the overparameterized regime -- unlike that of ridge regression or the lasso -- remains comparatively less explored. We contribute to this growing literature by deriving new algebraic and statistical results for the minimum $\ell_2$-norm OLS interpolator. In contrast to much of the existing work, which focuses on prediction risk, we center our analysis on parameter estimation and inference, which are fundamental for many statistics and causal inference applications. Specifically, we establish overparameterized analogues of (i) the leave-$k$-out formulas, (ii) the omitted variable bias formula, and (iii) the Frisch-Waugh-Lovell theorem. Under the Gauss-Markov model, we further extend the Gauss-Markov theorem and analyze variance estimation under homoskedasticity in the overparameterized setting. Collectively, these results provide a systematic framework for studying parameter estimation and inference in overparameterized linear models, offering a novel perspective on benign overfitting beyond its implications for prediction.

Conformal prediction provides distribution-free prediction intervals with finite-sample coverage guarantees, and recent work by Snell \& Griffiths reframes it as Bayesian Quadrature (BQ-CP), yielding powerful data-conditional guarantees via Dirichlet posteriors over thresholds. However, BQ-CP fundamentally requires the i.i.d. assumption. Meanwhile, weighted conformal prediction handles distribution shift via importance weights but remains frequentist, producing only point-estimate thresholds. We propose \textbf{Weighted Bayesian Conformal Prediction (WBCP)}, which generalizes BQ-CP to arbitrary importance-weighted settings by replacing the uniform Dirichlet $\Dir(1,\ldots,1)$ with a weighted Dirichlet $\Dir(\neff \cdot \tilde{w}_1, \ldots, \neff \cdot \tilde{w}_n)$, where $\neff$ is Kish's effective sample size. We prove four theoretical results: (1)~$\neff$ is the unique concentration parameter matching frequentist and Bayesian variances; (2)~posterior standard deviation decays as $O(1/\sqrt{\neff})$; (3)~BQ-CP's stochastic dominance guarantee extends to per-weight-profile data-conditional guarantees; (4)~the HPD threshold provides $O(1/\sqrt{\neff})$ improvement in conditional coverage. We instantiate WBCP for spatial prediction as \emph{Geographical BQ-CP}, where kernel-based spatial weights yield per-location posteriors with interpretable diagnostics. Experiments on synthetic and real-world spatial datasets demonstrate that WBCP maintains coverage guarantees while providing substantially richer uncertainty information.

In this paper, we propose an invariant quantile regression (IQR) framework specifically designed for multi-environment datasets, which captures the invariance across different environments. This framework is closely related to transfer learning, causal inference, and fair machine learning, and is motivated by scenarios in which the conditional probability of the response given covariates varies, while certain key variables remain invariant. This perspective differs notably from previous works that restrict attention to the conditional mean, which is often insufficient to capture the full causal relationships between covariates and the response in heterogeneous environments. In contrast, quantile-based invariance naturally accommodates heterogeneity, and aligns more closely with structural causal models, in which variables invariant across environments at one or multiple quantile levels directly indicate potential and stable causal variables. Moreover, we show that IQR may yield a larger set of endogenous variables compared to the conditional mean framework, which in turn promotes more effective exclusion of spurious (non-causal) variables. To achieve this, we introduce a Kernel-Smoothed Invariant Quantile Regression (KS-IQR) estimator, which leverages the underlying invariance structure and heterogeneity among environments, ensuring stable estimation across multiple environments. We establish the causal discovery properties of our method, demonstrate its ability to overcome the ``curse of endogeneity'', and derive an $\ell_2$ error bound for our estimator, all in a non-asymptotic framework. We apply our method to real data for causal discovery and obtain biologically meaningful relationships, recovering known signaling pathways and revealing additional quantile-specific effects.

For many health conditions, there are highly efficacious treatment and prevention products. Maximizing their impact requires strategies that improve the reach of health screening in order to establish who could benefit. For example, HIV prevention strategies aim to expand risk screening and to improve uptake of pre-exposure prophylaxis (PrEP) among those experiencing risk. Often, these strategies induce changes at the group-level (e.g., health clinics or communities) and are evaluated through cluster randomized trials. This scenario creates a complex, multilevel-mediation-missing data problem for the following reasons. First, the strategy is delivered at the cluster-level, while health screening and outcomes are at the individual-level. Second, the strategy improves health outcomes directly and indirectly through improved health screening. Third, everyone has an underlying status, which is only observed among those screened. To formally define the total effect in such settings, we use Counterfactual Strata Effects: causal estimands where the outcome is only relevant for a group whose membership is subject to missingness and/ or impacted by the exposure of interest. To identify and estimate the corresponding statistical estimand, we propose a novel extension of Two-Stage targeted minimum loss-based estimation (TMLE). Simulations demonstrate the practical performance of our approach as well as the limitations of existing approaches.

Whether a variable is the cause of another, or simply associated with it, is often an important scientific question. Causal Inference is the name associated with the body of techniques for addressing that question in a statistical setting. Although assessing causality is relatively straightforward in the presence of temporal information, outside of that setting - the situation considered here - it is more difficult to assess causal effects. The development of the field of causal inference has involved concepts from a wide range of topics, thereby limiting its adoption across some fields, including meteorology. However, at its core, the requisite knowledge for causal inference involves little more than basic probability theory and regression, topics familiar to most meteorologists. By focusing on these core areas, this and a companion article provide a steppingstone for the meteorology community into the field of (non-temporal) causal inference. Although some theoretical foundations are presented, the main goal is the application of a specific method, called matching, to a problem in meteorology. The data for the application are in public domain, and R code is provided as well, forming an easy path for meteorology students and researchers to enter the field.

One obstacle to ``elevating'' correlation to causation is the phenomenon of confounding, i.e., when a correlation between two variables exists because both variables are in fact caused by a third variable, called a confounder. The situation where the confounders are measured is examined in an earlier, accompanying article. Here, it is shown that even when the confounding variables are not measured, under certain conditions it is still possible to estimate the causal effect via a regression-based method that uses the notion of instrumental variables. Using a meteorological data set, similar to that in the sister article, a number of different estimates of the causal effect are compared and contrasted. It is shown that the instrumental-variable estimates of causal effect depend on the choice of the instrumental variable, and that meteorological considerations are important in resolving the ambiguity. R code is provided for generating all of the results, and numerous directions for future work are outlined.

When estimating causal effects from observational studies, researchers often need to adjust for many covariates to deconfound the non-causal relationship between exposure and outcome, among which many covariates are discrete. The behavior of commonly used estimators in the presence of many discrete covariates is not well understood since their properties are often analyzed under structural assumptions including sparsity and smoothness, which do not apply in discrete settings. In this work, we study the estimation of causal effects in a model where the covariates required for confounding adjustment are discrete but high-dimensional, meaning the number of categories $d$ is comparable with or even larger than sample size $n$. Specifically, we show the mean squared error of commonly used regression, weighting and doubly robust estimators is bounded by $\frac{d^2}{n^2}+\frac{1}{n}$. We then prove the minimax lower bound for the average treatment effect is of order $\frac{d^2}{n^2 \log^2 n}+\frac{1}{n}$, which characterizes the fundamental difficulty of causal effect estimation in the high-dimensional discrete setting, and shows the estimators mentioned above are rate-optimal up to log-factors. We further consider additional structures that can be exploited, namely effect homogeneity and prior knowledge of the covariate distribution, and propose new estimators that enjoy faster convergence rates of order $\frac{d}{n^2} + \frac{1}{n}$, which achieve consistency in a broader regime. The results are illustrated empirically via simulation studies.

Environmental problems are receiving increasing attention in socio-economic and health studies, fostering advances in recording and data collection of related real-life processes. However, traditional tools for data processing are often found too restrictive as they do not account for the rich nature of such data sets. In this paper, we propose a simple statistical perspective on forecasting environmental data collected sequentially over time across some predefined geographic region. We treat such data set as a surface (or functional) time series with a possibly complicated geographical domain. Using techniques from functional data analysis, we develop a forecasting methodology that allows to account for both geographic and temporal dependencies. This methodology allows integration of traditional multivariate techniques to provide forecasts surfaces. We demonstrate the practical value of our approach with a forecasting example of ground-level ozone concentration across Germany, showcasing its effectiveness and potential for broad application.

In Bayesian mixture models and other exchangeable-component models, the posterior is invariant under permutation of component labels, creating m! equivalent modes-the label-switching problem. Standard MCMC methods either mix poorly across these modes or rely on post-hoc relabelling that cannot guarantee the sampler has converged. We propose Folded Transport MCMC (FolT-MCMC), which eliminates label switching before sampling by restricting the Markov chain to a fundamental domain-a sorted or reflected subspace containing exactly one representative from each symmetric mode. The proposal is a learned normalising flow whose density is symmetrised over the group orbits, ensuring correct targeting on the reduced space. We show that this construction preserves a computable convergence diagnostic based on the oscillation of the log-density ratio, and that the diagnostic becomes sharper on the fundamental domain whenever the original-space flow under-covers one or more symmetric modes. Experiments on Gaussian mixtures (d=2-20), label-switching targets (up to 24 equivalent modes), a standard Bayesian three-component mixture posterior, and real accelerometer data from a supertall building show improvement ratios of 2x to 145x, with the folded diagnostic stable across dimensions while the unfolded diagnostic collapses.

Designing clinical trials requires evaluating multiple operating characteristics (OCs), such as the likelihood of an early stopping decision, the probability of detecting a treatment effect, and the Type I error rate. In most cases, these evaluations are based on computationally intensive Monte Carlo simulations. As the complexity of clinical trials and the use of adaptive designs increase, the computational burden can quickly become prohibitive. We introduce a strategy for rapidly approximating OCs, called the Q-approximation. Our approach is based on quadratic approximations of the log-likelihood and asymptotic arguments. The main idea is to replace simulation of full trial datasets with simulation of the approximate likelihood functions that determine the trial's interim and final decisions. The Q-approximation approach can be applied to any trial design that uses data analysis methods coherent with the likelihood principle, including multistage designs with early stopping, adaptively randomized designs, and designs that leverage external data. We illustrate the approach with several examples and show that it provides an accurate approximation of important OCs while reducing the computation time compared to Monte Carlo simulations. In particular, in our experiments, the standard Monte Carlo approximation of OCs requires 150 to 1,900 times greater computing budget than Q-approximations to achieve comparable levels of accuracy. By enabling fast OC evaluations, Q-approximations can support the broader use of innovative trial designs in both applied trial planning and methodological development.

Constructing accurate and computationally efficient surrogate models (or emulators) for predicting dynamical system responses is critical in many engineering domains, yet remains challenging due to the strongly nonlinear and high-dimensional mapping from external excitations and system parameters to system responses. This work introduces a novel Function-on-Function Nonlinear AutoRegressive model with eXogenous inputs (F2NARX), which reformulates the recently proposed $\mathcal{F}$-NARX method from a function-on-function regression perspective. The proposed framework substantially improves predictive efficiency while maintaining high accuracy. By combining principal component analysis with Gaussian process regression, F2NARX further enables probabilistic predictions of dynamical responses via the unscented transform in an autoregressive manner. Such probabilistic prediction capabilities further facilitate active learning for first-passage probability evaluation. The effectiveness of the method is demonstrated through case studies of varying complexity. Results show that F2NARX outperforms state-of-the-art NARX model by orders of magnitude in efficiency while achieving higher accuracy in general. Meanwhile, the active learning approach enables accurate estimation of first-passage failure probabilities for dynamical systems using only a small number of training time histories.

When a hidden Markov model permits the conditional likelihood of an observation given the hidden process to be zero, all particle simulations from one observation time to the next could produce zeros. If so, the filtering distribution cannot be estimated and the estimated parameter likelihood is zero. The alive particle filter addresses this by simulating a random number of particles for each inter-observation interval, stopping after a target number of non-zero conditional likelihoods. For outlying observations or poor parameter values, a non-zero result can be extremely unlikely, and computational costs prohibitive. We introduce the Frankenfilter, a principled, partially alive particle filter that targets a user-defined amount of success whilst fixing lower and upper bounds on the number of simulations. The Frankenfilter produces unbiased estimators of the likelihood, suitable for pseudo-marginal Metropolis--Hastings (PMMH). We demonstrate that PMMH with the Frankenfilter is more robust to outliers and mis-specified initial parameter values than PMMH using standard particle filters, and is typically at least 2-3 times more efficient. We also provide advice for choosing the amount of success. In the case of n exact observations, this is particularly simple: target n successes.

We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \in \mathbb{R}^{m \times n}$ and a factorization rank $r \ll \min(m, n)$, NMD seeks matrices $W \in \mathbb{R}^{m \times r}$ and $H \in \mathbb{R}^{r \times n}$ such that $X \approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \min(b, \max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications.

Data assimilation (DA) estimates a dynamical system's state from noisy observations. Recent generative models like the ensemble score filter (EnSF) improve DA in high-dimensional nonlinear settings but are computationally expensive. We introduce the ensemble flow filter (EnFF), a training-free, flow matching (FM)-based framework that accelerates sampling and offers flexibility in flow design. EnFF uses Monte Carlo estimators for the marginal flow field, localized guidance for observation assimilation, and utilizes a novel flow path that exploits the Bayesian DA formulation. It generalizes classical filters such as the bootstrap particle filter and ensemble Kalman filter. Experiments on high-dimensional benchmarks demonstrate EnFF's improved cost-accuracy tradeoffs and scalability, highlighting FM's potential for efficient, scalable DA. Code is available at https://github.com/Utah-Math-Data-Science/Data-Assimilation-Flow-Matching.

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

This manuscript contains preprint of a chapter under consideration for inclusion in the forthcoming third edition of {\em Cover and Thomas's Elements of Information Theory}, posted with permission from Wiley. The table of contents EIT-3 ToC of the new edition can be found at: https://docs.google.com/document/d/1L-m4oQEJw1PJhoxBeMwrrBD8S_HmvzMEkPbYvS24980/edit?usp=sharing . For feedback, please contact abbas@ee.stanford.edu Learning and information theory intersect in both model training and the characterization of fundamental performance limits. This manuscript provides a concise and accessible treatment of the first intersection, requiring only basic background in information theory and statistics at the senior undergraduate or first-year graduate level. End-of-chapter exercises make the material well suited for classroom use as well as self-study. The chapter focuses on the role of divergence measures in model training, with examples ranging from linear and logistic regression to autoregressive models, variational autoencoders, diffusion models, generative adversarial networks, and score-based models. It introduces the evidence lower bound (ELBO), f-divergences, and the Fisher divergence. In particular, the treatment of the generative diffusion model provides a more systematic and explicit derivation than is typical in the literature.

We study attention mechanisms through the lens of a canonical unsupervised problem: principal component analysis (PCA). We show that, when trained on Gaussian data, both softmax and linear attention layers learn parameters that align with the principal eigenvectors of the covariance matrix, thereby establishing a direct and explicit connection with PCA. Our analysis covers both finite and infinite prompt regimes. In the infinite-prompt limit, we prove convergence to globally optimal solutions aligned with the leading spectral direction, while in the finiteprompt setting we show that the same behavior emerges up to sampling effects. We further extend the analysis to an in-context setting with spiked Wishart covariances, where attention successfully recovers the underlying signal direction. These results demonstrate that attention inherently performs PCA-like computations under unsupervised objectives, providing a theoretical foundation for its representation-learning capabilities.

Chaos arises in many complex dynamical systems, from weather to power grids, but is difficult to accurately model with data-driven methods such as machine learning emulators. While emulators are promising tools for accelerating simulations and solving inverse problems, they still struggle to learn chaotic dynamics, where sensitivity to initial conditions renders exact long-term forecasts infeasible, especially given noisy data. Recent work instead trains emulators to match the statistical properties of chaotic attractors, but these approaches often rely on handcrafted summary statistics or large, diverse multi-environment datasets. In this work, we propose a family of adversarial optimal transport objectives that can jointly learn high-quality summary statistics and a physically consistent emulator from a single noisy trajectory. We theoretically analyze and experimentally validate a Sinkhorn divergence formulation (2-Wasserstein) and a WGAN-style dual formulation (1-Wasserstein) of our approach. Numerical experiments across a variety of chaotic systems, including ones with high-dimensional spatiotemporal chaos, show that emulators trained using our proposed objectives have significantly improved long-term statistical fidelity.

We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min-Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min-max characterization of key statistics, enabling approximation of the mean $μ_{\hatθ}$ and covariance $C_{\hatθ}$ of the ERM estimator $\hatθ$. Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training data, the projection $\hatθ^\top x$ approximately follows the convolution of the generally non-Gaussian distribution of $μ_{\hatθ}^\top x$ with an independent centered Gaussian variable of variance $\mathrm{tr}(C_{\hatθ} \mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $μ_{\hatθ}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights.

Statistical inference with bandit data presents fundamental challenges owing to adaptive sampling, which violates the independence assumptions underlying classical asymptotic theory. Recent work has identified stability~\citep{laiwei82} as a sufficient condition for valid inference under adaptivity. This paper first provides a refined stability condition, stated in terms of the iterates of an online algorithm, and shows that a large class of regularized stochastic-mirror-descent-style algorithms satisfy it. This refined condition allows us to strengthen the asymptotic results of~\citet{laiwei82} in several ways. First, we derive a non-asymptotic Berry--Esseen bound for the empirical reward estimates under adaptive sampling. Second, we derive matching non-asymptotic upper and lower bounds on the regret of the proposed algorithm, yielding a precise characterization of its regret. Third, we show that these regularized algorithms preserve asymptotic normality and valid inference under a prescribed level of adversarial corruption. Finally, we show that regularization is necessary rather than incidental: Lai--Wei stability is incompatible with the optimal $O(\sqrt{T})$ regret rate -- the rate attained by unregularized algorithms such as EXP3 -- so that a controlled, polylogarithmic inflation in regret is the price of valid inference.

Controlling generative models is computationally expensive. This is because optimal alignment with a reward function--whether via inference-time steering or fine-tuning--requires estimating the value function. This task demands access to the conditional posterior $p_{1|t}(x_1|x_t)$, the distribution of clean data $x_1$ consistent with an intermediate state $x_t$, a requirement that typically compels methods to resort to costly trajectory simulations. To address this bottleneck, we introduce Meta Flow Maps (MFMs), a framework extending consistency models and flow maps into the stochastic regime. MFMs are trained to perform stochastic one-step posterior sampling, generating arbitrarily many i.i.d. draws of clean data $x_1$ from any intermediate state. Crucially, these samples provide a differentiable reparametrization that unlocks efficient value function estimation. We leverage this capability to solve bottlenecks in both paradigms: enabling inference-time steering without inner rollouts, and facilitating unbiased, off-policy fine-tuning to general rewards. Empirically, our single-particle steered-MFM sampler outperforms a Best-of-1000 baseline on ImageNet across multiple rewards at a fraction of the compute.

Predictions from statistical physics postulate that recovery of the communities in the Stochastic Block Model (SBM) with a fixed number $K$ of communities is possible in polynomial time above, and only above, the Kesten-Stigum (KS) threshold. This conjecture has given rise to a rich literature, proving that non-trivial community recovery is indeed possible in SBM above the KS threshold. Failure of low-degree polynomials (LDP) below the KS threshold was also proven, as long as $K\ll \sqrt{n}$, where $n$ is the number of nodes in the observed graph. When $K\geq \sqrt{n}$, Chin et al.(2025) recently proved that, in a \emph{sparse regime}, community recovery in polynomial time is possible below the KS threshold by counting non-backtracking paths. This breakthrough led them to postulate a new threshold for the many-communities regime $K\geq \sqrt{n}$. In this work, we provide evidence supporting their conjecture:\\ 1- We prove that, for \emph{any graph density}, LDP fail to recover communities below the threshold postulated by Chin et al.(2025) ;\\ 2- We prove that community recovery is possible in polynomial time above the postulated threshold, not only in the \emph{sparse regime} considered in Chin et al.~(2025), but also in \emph{moderately sparse regimes}, by counting occurrences of some specific motifs inspired by the LDP analysis.\\ In particular, counting self-avoiding paths of length $\log(n)$, which is closely related to spectral algorithms based on the Non-Backtracking operator, is optimal only in the sparse regime. More complex motifs based on the blow-up of a cycle must be considered in denser regimes.

Unstructured text provides decision-makers with a rich data source in many domains, ranging from product reviews in retail to nursing notes in healthcare. To leverage this information, words are typically translated into word embeddings -- vectors that encode the semantic relationships between words -- through unsupervised learning algorithms such as matrix factorization. However, learning word embeddings from new domains with limited training data can be challenging, because the meaning/usage may be different in the new domain, e.g., the word ``positive'' typically has positive sentiment, but often has negative sentiment in medical notes since it may imply that a patient tested positive for a disease. In practice, we expect that only a small number of domain-specific words may have new meanings. We propose an intuitive two-stage estimator that exploits this structure via a group-sparse penalty to efficiently transfer learn domain-specific word embeddings by combining large-scale text corpora (such as Wikipedia) with limited domain-specific text data. We bound the generalization error of our transfer learning estimator, proving that it can achieve high accuracy with substantially less domain-specific data when only a small number of embeddings are altered between domains. Furthermore, we prove that all local minima identified by our nonconvex objective function are statistically indistinguishable from the global minimum under standard regularization conditions, implying that our estimator can be computed efficiently. Our results provide the first bounds on group-sparse matrix factorization, which may be of independent interest. We empirically evaluate our approach compared to state-of-the-art fine-tuning heuristics from natural language processing.

Ongoing advancements in microbiome profiling have provided unprecedented insights into the molecular dynamics of microbial communities, sparking a surge of interest in uncovering the microbiome's critical role in human health. Identifying microbial features linked to clinical outcomes, however, remains challenging due to the high-dimensional, sparse, and compositional nature of microbiome data. Additionally, many microbial taxa, although classified as distinct, may share functional roles, complicating traditional variable selection methods. To overcome these obstacles, we introduce Bayesian Regression with Agglomerated Compositional Effects (BRACE), a novel approach using a spike-and-cluster prior combining Bernoulli activity indicators, an Ewens exchangeable partition prior on the finite active set, and a projection-based constrained Gaussian prior on cluster effects to perform data-adaptive clustering and variable selection. The methodological innovation of our work lies in how we combine the Ewens partition prior with a projection-based constrained Gaussian on the cluster atoms to enforce the sum-to-zero constraint. BRACE groups microbial taxa with similar effects on the outcome, yielding more interpretable models while enabling effective dimension reduction. Through comprehensive simulations and a real-world application examining the influence of oral microbiome composition on insulin resistance, we demonstrate BRACE's superior performance over existing methods, particularly in identifying key features with shared effects on outcomes.

The correct inferential object in claims reserving is the conditional predictive distribution $p(R \mid \mathcal{D}, \hatθ)$, where $\mathcal{D}$ is the observed triangle held fixed. We refer to this as the conditioning principle. All existing bootstraps violate it by resampling functions of $\mathcal{D}$ inside the predictive loop, producing an $O(1)$ coverage error that does not vanish as the triangle grows. The Dirichlet-Gamma hierarchy admits a bootstrap that satisfies the principle exactly: $S^{IBNP}_i = X^{obs}_i (1-W_i)/W_i$ with $W_i \sim \mathrm{Beta}(c\hat{F}_{I-i}, c(1-\hat{F}_{I-i}))$ sampled directly from its predictive distribution. Only the allocation proportion $W_i$ is simulated; the observed triangle is held fixed. It thus inherits calibration from any development-proportion method (Chain-Ladder, Bornhuetter-Ferguson, Cape Cod, or other), making it model-agnostic. The coverage deficit is $O(I^{-1/2})$, independent of the number of development periods. Under compound Poisson data-generating processes the bootstrap is conservative for every $F_{I-i} \in (0,1)$: the predictive standard deviation analytically exceeds the true value by the factor $1/\sqrt{F_{I-i}}$. The ODP bootstrap violates the principle through two mechanisms in opposite directions: re-estimation inflates bootstrap variance under the ODP DGP, while missing accident-year frailty deflates it under frailty DGPs. The resulting coverage discrepancy is $Ω(1)$ regardless of $I$, providing a structural explanation for the cross-portfolio miscalibration heterogeneity documented by Meyers (2015). Chain-Ladder, Bornhuetter-Ferguson and Cape Cod emerge as credibility estimators under diffuse, informative and pooling priors respectively, with identical structure for counts and amounts. The concentration $c$ serves as a diagnostic: $\hat{c} < 30$ signals non-stationary development.

The Chain-Ladder (CL) method remains the dominant macro-level technique for claims reserving in non-life insurance, yet its classical formulation lacks a coherent probabilistic foundation. Existing stochastic extensions-including the Mack model and the Over-Dispersed Poisson (ODP) framework-provide measures of uncertainty but rely on second-moment assumptions or quasi-likelihood variance structures without clear generative interpretations. This paper develops a Negative Binomial Chain-Ladder (NB-CL) model that embeds the CL method within a full likelihood-based framework. The key contribution is a micro-level derivation showing that the negative binomial distribution arises naturally from a Poisson-Gamma construction: claims arrive according to a Poisson process with Gamma-distributed accident-year heterogeneity, and aggregation yields negative binomial incremental counts. This derivation gives the dispersion parameter $κ$ a structural interpretation as accident-year heterogeneity, rather than an ad-hoc overdispersion adjustment. The NB-CL model generalises the Poisson Chain-Ladder model in the limit $κ\to \infty$, shares the point estimates of the ODP model while differing in its variance function (quadratic vs. linear), and unifies the Chain-Ladder family within a single probabilistic hierarchy. A parametric bootstrap procedure is developed to incorporate both process and parameter uncertainty. Simulation studies confirm near-nominal coverage under correct specification once the dispersion parameter is bias-corrected, and a controlled degradation under model misspecification. Empirical illustrations on claim count data (Australian motor bodily injury) and paid amounts (Taylor-Ashe) document both the structural reading of $κ$ and the working-approximation status of the model in the amounts case.

This paper investigates the impact of carbon pricing under the EU Emissions Trading System (EU ETS) on the Italian electricity market, focusing on the carbon cost pass-through rate (CPTR) across market zones during Phases 3 and 4 (2016-2024). Using daily data, the study applies an econometric framework based on a linear regression model with autoregressive dynamics to estimate the extent to which carbon costs are reflected in wholesale electricity prices. It further incorporates robustness checks and quantile regression to assess how the CPTR varies across different fuel spread levels. The results show that carbon costs are positively and significantly transmitted to electricity prices, confirming the relevance of carbon pricing as a key market driver. However, pass-through is incomplete, with CPTR values consistently below 100%. At the national level, the pass-through estimate is around 32%, with no statistically significant change between Phase 3 and Phase 4. Substantial heterogeneity emerges across market zones: pass-through increases in the North, Centre-North, and Sardinia during Phase 4, while it declines in the Centre-South and Sicily, reflecting differences in generation mix, carbon intensity, and market conditions. Overall, the findings highlight the importance of market zones factors in shaping the effectiveness of carbon pricing in electricity markets.

A utility function has been proposed that values more lives that are lost than those that are saved. I do not dispute the ethical motivation behind this kind of asymmetry. However, I show with a simple example that the scope of applicability of such a decision criterion is considerably more limited than it may first appear.

The Swiss system is an increasingly popular competition format as it provides a favourable trade-off between the number of matches and ranking accuracy. However, there is no empirical study on the potential unfairness of Swiss-system chess tournaments if an odd number of rounds is played. To analyse this issue, our paper compares the number of points scored in the tournament between players who played one game more with the white pieces and players who played one game fewer with the white pieces. Using data from 28 highly prestigious competitions, we find that players with an extra white game score significantly more points. In particular, the advantage exceeds the value of a draw in the four Grand Swiss tournaments. A potential solution to this unfairness could be organising Swiss-system chess tournaments with an even number of rounds, and guaranteeing a balanced colour assignment for all players using a recently proposed pairing mechanism.

We propose a new method to estimate structural parameters in multi-way networks while controlling for rich structures of fixed effects. The method is based on a series of classification tasks and is agnostic to both the number and structure of fixed effects. In contrast to full maximum likelihood approaches, our estimator does not suffer from the incidental parameter problem. For sparsely connected networks, it is also computationally faster than PPML. We provide empirical evidence that our estimator yields more reliable confidence intervals than PPML and its bias-correction strategies. These improvements hold even under model misspecification and are more pronounced in sparse settings. While PPML remains competitive in dense, low-dimensional data, our approach offers a robust alternative for multi-way models that scales efficiently with sparsity. The method is applied to study the causal effect of a policy reform on spatial accessibility to health care in France.

A common approach to out-of-distribution prediction restricts models to causal or invariant covariates to avoid spurious associations that may change across environments. Despite its theoretical appeal, this strategy can underperform empirical risk minimization when only a subset of the causal parents of the outcome is observed. In such settings, non-causal covariates can serve as proxies for unobserved causal parents and improve prediction when the proxy relationship is stable, but they can hurt when shifts disrupt that relationship. Thus, the optimal covariate set can depend on the specific shift encountered. Because different shifts leave signatures in the unlabeled covariate distribution, we propose an environment-adaptive covariate selection algorithm that maps environment-level summaries to environment-specific covariate sets. These summaries may be hand-crafted or learned from multi-environment data, and prior causal knowledge can be incorporated as constraints. Across simulations and applied datasets, the proposed method improves over static causal, invariant, and other non-adaptive rules under diverse shifts.

As data science emerges as a distinct academic discipline, introductory data science (IDS) courses play a key role in shaping students foundational understanding. Often taught by instructors without formal training in data science or pedagogy, these courses present a unique and globally relevant context for examining pedagogical content knowledge (PCK). Drawing on semi-structured interviews with 14 IDS instructors and their course syllabi, this study explores how IDS instructors describe and make sense of their teaching practices, which are analyzed through the lens of PCK. The findings highlight key components of PCK about IDS and offer insights into supporting instructor development. This work contributes to expanding the scope of PCK research into new interdisciplinary domains and ongoing global efforts to build capacity in data science education. It could serve as a starting point for developing a PCK framework specific to IDS.

The drastic changes in the global economy, geopolitical conditions, and disruptions such as the COVID-19 pandemic have impacted the cost of living and quality of life. It is essential to comprehend the long-term implications of the cost of living and quality of life in major economies. A transparent and comprehensive living index must include multiple dimensions of living conditions. In this study, we present an approach to quantifying the quality of life through the Global Ease of Living Index that combines various socio-economic and infrastructural factors into a single composite score. Our index utilises economic indicators that define living standards, which could help in targeted interventions to improve specific areas. We present a machine learning framework to address missing data for certain economic indicators in specific countries. We then curate and update the data and use a dimensionality reduction approach (Principal Component Analysis and Factor Analysis) to create the Ease of Living Index for major economies since 1970. Our work significantly adds to the literature by offering a practical tool for policymakers to identify areas needing improvement, such as healthcare systems, employment opportunities, and public safety. Our approach with open data and code can be easily reproduced and applied to various contexts, providing transparency and accessibility for ongoing research and policy development in quality-of-life assessment.

The expected signature uniquely determines the law of a random rough path under a moment-growth condition, yet finite-sample bounds for estimating its truncations from a single long dependent trajectory remain unavailable. We study a strictly stationary stochastic process equipped with a geometric rough-path lift, observed in non-overlapping blocks of equally-spaced samples, and prove a non-asymptotic mean-squared error (MSE) bound for the block-averaging estimator of its truncated expected signature. Under moment and stationarity assumptions together with a direct covariance-decay condition on block signatures -- strictly weaker than $α$-mixing and applicable to long-range-dependent processes -- the error separates into a discretization term and a fluctuation term, with rates determined respectively by path regularity and dependence strength. A levelwise rough-factorial variance analysis keeps finite-truncation constants explicit and yields an optimal allocation rule under a fixed observation budget. We verify the assumptions for independent-coordinate fractional Ornstein--Uhlenbeck processes in three regimes: short-range (Hurst $1/4<H<1/2$), semimartingale ($H=1/2$), and long-range ($H>1/2$); in all three, the block-signature covariance is summable, so the fluctuation term decays at the same rate as in the independent-block case, even under long memory at $H>1/2$. Monte Carlo experiments show empirical slopes steeper than the guaranteed upper-bound rates.

The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different families of functions $f$. The article establishes convergence of the power series of $f(B)$ and $f(T)$ under the operator norm for the Wiener algebra $\mathbb{W}_+$, and convergence under strong operator topology for $f$ in $H^{\infty}$, based on the use of Abel sums. Based on this calculus, it unifies the notion of stationary process invertibility with the operator invertibility of the transfer function $f(T)$.

Standard approaches to quantum statistical inference rely on measurements that induce a collapse of the wave function, effectively consuming the quantum state to extract information. In this work, we investigate the fundamental limits of \emph{locally-gentle} quantum state certification, where the learning algorithm is constrained to perturb the state by at most $α$ in trace norm, thereby allowing for the reuse of samples. We analyze the hypothesis testing problem of distinguishing whether an unknown state $ρ$ is equal to a reference $ρ_0$ or $ε$-far from it. We derive the minimax sample complexity for this problem, quantifying the information-theoretic price of non-destructive measurements. Specifically, by constructing explicit measurement operators, we show that the constraint of $α$-gentleness imposes a sample size penalty of $\frac{d}{α^2}$, yielding a total sample complexity of $n = Θ(\frac{d^3}{ε^2 α^2})$. Our results clarify the trade-off between information extraction and state disturbance, and highlight deep connections between physical measurement constraints and privacy mechanisms in quantum learning. Crucially, we find that the sample size penalty incurred by enforcing $α$-gentleness scales linearly with the Hilbert-space dimension $d$ rather than the number of parameters $d^2-1$ typical for high-dimensional private estimation.

Statistical literature provides pointwise limiting distributions of the nonparametric maximum likelihood estimator (NPMLE) in monotone binary regression for the two extremal cases: If the feature-label relation is strictly monotone and sufficiently smooth, it converges at a cube-root-$n$ rate with scaled Chernoff-type limiting distribution, and it converges at the parametric $\sqrt{n}$-rate if the underlying relation is flat. In this article, we provide the complete picture of the distributional metamorphosis of the NPMLE, revealing a new limiting distribution which provides a significantly better distributional approximation for small samples in case of a weak feature-label relationship. It is shown to continuously interpolate between the two extremal cases. The innovative way to determine this distribution is to generate it as a limit of the NPMLE in the newly introduced weak-feature-impact triangular array for a particular parameter-sample-size constellation. Moreover, the phase transition is likewise observed for the suitably rescaled $L^{1}$-error in this weak-feature-impact scenario. As a by-product, its limiting distribution for flat regression functions is obtained, which was unknown before. The proof develops a completely new strategy, notably not based on the switch relation. A novel type of local minimax lower bounds accompanies these results.