Papers

The scenario approach is an established data-driven design framework that comes equipped with a powerful theory linking design complexity to generalization properties. In this approach, data are simultaneously used both for design and for certifying the design's reliability, without resorting to a separate test dataset. This paper takes a step further by guaranteeing additional properties, useful in post-design usage but not considered during the design phase. To this end, we introduce a two-level framework of appropriateness: baseline appropriateness, which guides the design process, and post-design appropriateness, which serves as a criterion for a posteriori evaluation. We provide distribution-free upper bounds on the risk of failing to meet the post-design appropriateness; these bounds are computable without using any additional test data. Under additional assumptions, lower bounds are also derived. As part of an effort to demonstrate the usefulness of the proposed methodology, the paper presents two practical examples in H2 and pole-placement problems. Moreover, a method is provided to infer comprehensive distributional knowledge of relevant performance indexes from the available dataset.

In Neyman's original formulation, a 1-alpha confidence interval procedure is justified by its long-run coverage properties, and a single realized interval is to be described only by the slogan that it either covers the parameter or it does not. On this view, post-data probability statements about the coverage of an individual interval are taken to be conceptually out of bounds. In this paper, I present two kinds of arguments against treating that "either-or" reading as the only legitimate interpretation of confidence. The first is informal, via a set of thought experiments in which the same joint probability model is used to compute both forward-looking and backward-looking probabilities for occurred-but-unobserved events. The second is more formal, recasting the standard confidence-interval construction in terms of infinite sequences of trials and their associated 0/1 coverage indicators. In that representation, the design-level coverage probability 1-alpha and the degenerate conditional probabilities given the full data appear simply as different conditioning levels of the same model. I argue that a strict behavioristic reading that privileges only the latter is in tension with the very mathematical machinery used to define long-run error rates. I then sketch an alternative view of confidence as a predictive probability (or forecast) about the coverage indicator, together with a simple normative rule for when intermediate probabilities for single coverage events should be allowed. Keywords: confidence intervals; coverage probability; frequentist inference; single-case probability; predictive probability; Neyman. Disclaimer: The findings and conclusions in this report are those of the author and do not necessarily represent the official position of the Centers for Disease Control and Prevention.

Adaptive randomized experiments update treatment probabilities as data accrue, but still require an end-of-study interval for the average treatment effect (ATE) at a prespecified horizon. Under adaptive assignment, propensities can keep changing, so the predictable quadratic variation of AIPW/DML score increments may remain random. When no deterministic variance limit exists, Wald statistics normalized by a single long-run variance target can be conditionally miscalibrated given the realized variance regime. We assume no interference, sequential randomization, i.i.d. arrivals, and executed overlap on a prespecified scored set, and we require two auditable pipeline conditions: the platform logs the executed randomization probability for each unit, and the nuisance regressions used to score unit $t$ are constructed predictably from past data only. These conditions make the centered AIPW/DML scores an exact martingale difference sequence. Using self-normalized martingale limit theory, we show that the Studentized statistic, with variance estimated by realized quadratic variation, is asymptotically N(0,1) at the prespecified horizon, even without variance stabilization. Simulations validate the theory and highlight when standard fixed-variance Wald reporting fails.

We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled trajectory. Our result characterizes the long-term fluctuations of the algorithm around the minimizer by providing a diffusion limit for the trajectory. In contrast with classical central limit theorems for the last iterate or Polyak-Ruppert averages, this functional result captures the temporal structure of the fluctuations and applies to non-smooth settings such as robust location estimation, including the geometric median.

Stability and robustness are critical for deploying Transformers in safety-sensitive settings. A principled way to enforce such behavior is to constrain the model's Lipschitz constant. However, approximation-theoretic guarantees for architectures that explicitly preserve Lipschitz continuity have yet to be established. In this work, we bridge this gap by introducing a class of gradient-descent-type in-context Transformers that are Lipschitz-continuous by construction. We realize both MLP and attention blocks as explicit Euler steps of negative gradient flows, ensuring inherent stability without sacrificing expressivity. We prove a universal approximation theorem for this class within a Lipschitz-constrained function space. Crucially, our analysis adopts a measure-theoretic formalism, interpreting Transformers as operators on probability measures, to yield approximation guarantees independent of token count. These results provide a rigorous theoretical foundation for the design of robust, Lipschitz continuous Transformer architectures.

When using the Focused Information Criterion (FIC) for assessing and ranking candidate models with respect to how well they do for a given estimation task, it is customary to produce a so-called FIC plot. This plot has the different point estimates along the y-axis and the root-FIC scores on the x-axis, these being the estimated root-mean-square scores. In this paper we address the estimation uncertainty involved in each of the points of such a FIC plot. This needs careful assessment of each of the estimators from the candidate models, taking also modelling bias into account, along with the relative precision of the associated estimated mean squared error quantities. We use confidence distributions for these endeavours. This leads to fruitful CD-FIC plots, helping the statistician to judge to what extent the seemingly best models really are better than other models, etc. These efforts also lead to two further developments. The first is a new tool for model selection, which we call the quantile FIC, which helps overcome certain difficulties associated with the usual FIC procedures, related to somewhat arbitrary schemes for handling estimated squared biases. A particular case is the median-FIC. The second development is to form model averaged estimators with fruitful weights determined by the relative sizes of the median- and quantile-FIC scores. And Mrs. Jones is pregnant.

For a broad family of discriminative models that includes autoregressive language models, identifiability results imply that if two models induce the same conditional distributions, then their internal representations are equal up to an invertible linear transformation. We ask whether an analogous conclusion holds approximately when the distributions are close instead of equal. Building on the observation of Nielsen et al. (2025) that closeness in KL divergence need not imply high linear representational similarity, we study a distributional distance based on logit differences and show that closeness in this distance does yield linear similarity guarantees. Specifically, we define a representational dissimilarity measure based on the models' identifiability class and prove that it is bounded by the logit distance. We further show that, when model probabilities are bounded away from zero, KL divergence upper-bounds logit distance; yet the resulting bound fails to provide nontrivial control in practice. As a consequence, KL-based distillation can match a teacher's predictions while failing to preserve linear representational properties, such as linear-probe recoverability of human-interpretable concepts. In distillation experiments on synthetic and image datasets, logit-distance distillation yields students with higher linear representational similarity and better preservation of the teacher's linearly recoverable concepts.

We examine static and dynamic social network structure in 176 villages within the Copan Department of Honduras across two data waves (2016, 2019), using detailed data on multiplex networks for 20,232 individuals enrolled in a longitudinal survey. These networks capture friendship, health advice, financial help, and adversarial relationships, allowing us to show how cooperation and conflict jointly shape social structure. Using node-level network measures derived from near-census sociocentric village networks, we leverage mixed-effects zero-inflated negative binomial models to assess the influence of individual attributes, such as gender, marital status, education, religion, and indigenous status, and of village characteristics, on the dynamics of social networks over time. We complement these node-level models with dyadic assortativity (odds-ratio-based homophily) and community-level measures to describe how sorting by key attributes differs across network types and between waves. Our results demonstrate significant assortativity based on gender and religion, particularly within health and financial networks. Across networks, gender and religion exhibit the most consistent assortative mixing. Additionally, community-level assortativity metrics indicate that educational and financial factors increasingly influence social ties over time. Our findings provide insights into how personal attributes and community dynamics interact to shape network formation and socio-economic relationships in rural settings over time.

This paper investigates the construction of space-filling designs for computer experiments. The space-filling property is characterized by the covering and separation radii of a design, which are integrated through the unified criterion of quasi-uniformity. We focus on a special class of designs, known as quasi-Monte Carlo (QMC) lattice point sets, and propose two construction algorithms. The first algorithm generates rank-1 lattice point sets as an approximation of quasi-uniform Kronecker sequences, where the generating vector is determined explicitly. As a byproduct of our analysis, we prove that this explicit point set achieves an isotropic discrepancy of $O(N^{-1/d})$. The second algorithm utilizes Korobov lattice point sets, employing the Lenstra–Lenstra–Lovász (LLL) basis reduction algorithm to identify the generating vector that ensures quasi-uniformity. Numerical experiments are provided to validate our theoretical claims regarding quasi-uniformity. Furthermore, we conduct empirical comparisons between various QMC point sets in the context of Gaussian process regression, showcasing the efficacy of the proposed designs for computer experiments.

Gene-gene and gene-environment interactions are widely believed to play significant roles in explaining the variability of complex traits. While substantial research exists in this area, a comprehensive statistical framework that addresses multiple sources of uncertainty simultaneously remains lacking. In this article, we synthesize and propose extension of a novel class of Bayesian nonparametric approaches that account for interactions among genes, loci, and environmental factors while accommodating uncertainty about population substructure. Our contribution is threefold: (1) We provide a unified exposition of hierarchical Bayesian models driven by Dirichlet processes for genetic interactions, clarifying their conceptual advantages over traditional regression approaches; (2) We shed light on new computational strategies that combine transformation-based MCMC with parallel processing for scalable inference; and (3) We present enhanced hypothesis testing procedures for identifying disease-predisposing loci.Through applications to myocardial infarction data, we demonstrate how these methods offer biological insights not readily obtainable from standard approaches. Our synthesis highlights the advantages of Bayesian nonparametric thinking in genetic epidemiology while providing practical guidance for implementation.

The chain-ladder (CL) method is the most widely used claims reserving technique in non-life insurance. This manuscript introduces a novel approach to computing the CL reserves based on a fundamental restructuring of the data utilization for the CL prediction procedure. Instead of rolling forward the cumulative claims with estimated CL factors, we estimate multi-period factors that project the latest observations directly to the ultimate claims. This alternative perspective on CL reserving creates a natural pathway for the application of machine learning techniques to individual claims reserving. As a proof of concept, we present a small-scale real data application employing neural networks for individual claims reserving.

Longitudinal electronic health record (EHR) data offer opportunities to study biomarker trajectories; however, association estimates-the primary inferential target-from standard models designed for regular observation times may be biased by a two-stage hierarchical missingness mechanism. The first stage is the visiting process (informative presence), where encounters occur at irregular times driven by patient health status; the second is the observation process (informative observation), where biomarkers are selectively measured during visits. To address these mechanisms, we propose a unified semiparametric joint modeling framework that simultaneously characterizes the visiting, biomarker observation, and longitudinal outcome processes. Central to this framework is a shared subject-specific Gaussian latent variable that captures unmeasured frailty and induces dependence across all components. We develop a three-stage estimation procedure and establish the consistency and asymptotic normality of our estimators. We also introduce a sequential procedure that imputes missing biomarkers prior to adjusting for irregular visiting and examine its performance. Simulation results demonstrate that our method yields unbiased estimates under this mechanism, whereas existing approaches can be substantially biased; notably, methods adjusting only for irregular visiting may exhibit even greater bias than those ignoring both mechanisms. We apply our framework to data from the All of Us Research Program to investigate associations between neighborhood-level socioeconomic status indicators and six blood-based biomarker trajectories, providing a robust tool for outpatient settings where irregular monitoring and selective measurement are prevalent.

We introduce a novel class of non-stationary covariance functions for random fields on linear networks that allows both the variance and the correlation range of the random field to vary spatially. The proposed covariance functions are useful to model random fields with a spatial dependence that is locally isotropic with respect to the resistance metric, a distance that reflects the topology of the network. The framework admits explicit stochastic representations of the associated random fields and can be naturally extended to matrix-valued covariance functions for vector-valued random fields. We assess the statistical and computational performance of a weighted local likelihood estimator for the proposed models using synthetic data generated on the street network of the University of Chicago neighborhood.

Machine learning model performance improvements tend to arise from competition and application. For deployment, we consider prescriptive scaling laws: given a pre-training compute budget, what downstream accuracy is attainable with contemporary post-training practice, and how stable is that mapping as the field evolves? Using large-scale observational evaluations with 5k existing and 2k newly evaluated model checkpoints spanning 2022-2026 across six benchmarks, we estimate capability boundaries, high conditional quantiles of benchmark scores as a function of log pre-training FLOPs, via smoothed quantile regression with a monotone, saturating sigmoid parameterization. We validate temporal reliability by fitting on earlier model generations and evaluating on later releases: across four of six tasks, the out-of-distribution coverage error remains below 2%, while math reasoning exhibits a consistently advancing boundary over time. For instance, at a budget of 10^24 FLOPs, the estimated attainable accuracies are 0.83 on IFEval and 0.54 on MATH Lvl 5. We then extend our approach to analyze task-dependent saturation and to probe contamination-related shifts on math reasoning tasks. Finally, we introduce a balanced I-optimal sampling algorithm that recovers near-full-data frontiers using roughly 20% of the parameter-count-weighted evaluation budget, as low as 5% on some tasks, while maintaining comparable calibration. Together, our work releases Proteus-2k, the latest model performance evaluation dataset, and introduces a practical methodology for translating compute budgets into reliable performance expectations and for monitoring when capability boundaries shift across time.

We propose a Bayesian copula-based framework to quantify clinically interpretable joint tail risks from paired continuous biomarkers. After converting each biomarker margin to rank-based pseudo-observations, we model dependence using one-parameter Archimedean copulas and focus on three probability-scale summaries at tail level $\alpha$: the lower-tail joint risk $R_L(\theta)=C_\theta(\alpha,\alpha)$, the upper-tail joint risk $R_U(\theta)=2\alpha-1+C_\theta(1-\alpha,1-\alpha)$, and the conditional lower-tail risk $R_C(\theta)=R_L(\theta)/\alpha$. Uncertainty is quantified via a restricted Jeffreys prior on the copula parameter and grid-based posterior approximation, which induces an exact posterior for each tail-risk functional. In simulations from Clayton and Gumbel copulas across multiple dependence strengths, posterior credible intervals achieve near-nominal coverage for $R_L$, $R_U$, and $R_C$. We then analyze NHANES 2017–2018 fasting glucose (GLU) and HbA1c (GHB) ($n=2887$) at $\alpha=0.05$, obtaining tight posterior credible intervals for both the dependence parameter and induced tail risks. The results reveal markedly elevated extremal co-movement relative to independence; under the Gumbel model, the posterior mean joint upper-tail risk is $R_U(\alpha)=0.0286$, approximately $11.46\times$ the independence benchmark $\alpha^2=0.0025$. Overall, the proposed approach provides a principled, dependence-aware method for reporting joint and conditional extremal-risk summaries with Bayesian uncertainty quantification in biomedical applications.

Zero-shot anomaly detection (ZSAD) has gained increasing attention in medical imaging as a way to identify abnormalities without task-specific supervision, but most advances remain limited to 2D datasets. Extending ZSAD to 3D medical images has proven challenging, with existing methods relying on slice-wise features and vision-language models, which fail to capture volumetric structure. In this paper, we introduce a fully training-free framework for ZSAD in 3D brain MRI that constructs localized volumetric tokens by aggregating multi-axis slices processed by 2D foundation models. These 3D patch tokens restore cubic spatial context and integrate directly with distance-based, batch-level anomaly detection pipelines. The framework provides compact 3D representations that are practical to compute on standard GPUs and require no fine-tuning, prompts, or supervision. Our results show that training-free, batch-based ZSAD can be effectively extended from 2D encoders to full 3D MRI volumes, offering a simple and robust approach for volumetric anomaly detection.

Causal structure learning, also known as causal discovery, aims to estimate causal relationships between variables as a form of a causal directed acyclic graph (DAG) from observational data. One of the major frameworks is the order-based approach that first estimates a topological order of the underlying DAG and then prunes spurious edges from the fully-connected DAG induced by the estimated topological order. Previous studies often focus on the former ordering step because it can dramatically reduce the search space of DAGs. In practice, the latter pruning step is equally crucial for ensuring both computational efficiency and estimation accuracy. Most existing methods employ a pruning technique based on generalized additive models and hypothesis testing, commonly known as CAM-pruning. However, this approach can be a computational bottleneck as it requires repeatedly fitting additive models for all variables. Furthermore, it may harm estimation quality due to multiple testing. To address these issues, we introduce a new pruning method based on sparse additive models, which enables direct pruning of redundant edges without relying on hypothesis testing. We propose an efficient algorithm for learning sparse additive models by combining the randomized tree embedding technique with group-wise sparse regression. Experimental results on both synthetic and real datasets demonstrated that our method is significantly faster than existing pruning methods while maintaining comparable or superior accuracy.

Mixture distributions are a workhorse model for multimodal data in information theory, signal processing, and machine learning. Yet even when each component density is simple, the differential entropy of the mixture is notoriously hard to compute because the mixture couples a logarithm with a sum. This paper develops a deterministic, closed-form toolkit for bounding and accurately approximating mixture entropy directly from component parameters. Our starting point is an information-theoretic channel viewpoint: the latent mixture label plays the role of an input, and the observation is the output. This viewpoint separates mixture entropy into an average within-component uncertainty plus an overlap term that quantifies how much the observation reveals about the hidden label. We then bound and approximate this overlap term using pairwise overlap integrals between component densities, yielding explicit expressions whenever these overlaps admit a closed form. A simple, family-dependent offset corrects the systematic bias of the Jensen overlap bound and is calibrated to be exact in the two limiting regimes of complete overlap and near-perfect separation. A final clipping step guarantees that the estimate always respects universal information-theoretic bounds. Closed-form specializations are provided for Gaussian, factorized Laplacian, uniform, and hybrid mixtures, and numerical experiments validate the resulting bounds and approximations across separation, dimension, number of components, and correlated covariances.

We develop a unified framework for goodness-of-fit (GOF) testing through the lens of Bayes risk. Classical GOF procedures are commonly calibrated either at fixed significance level (CLT scale) or through exponential error exponents (LDP scale). We establish that Bayes-risk optimal calibration operates on the moderate-deviation (MDP) scale, producing canonical $\sqrt{\log n}$ inflation of rejection thresholds and polynomially decaying Type I error. Our main contributions are: (i) we formalise the Rubin–Sethuraman program for KS-type statistics as a risk-calibration theorem with explicit regularity conditions on priors and empirical-process functionals; (ii) we develop the precise connection between Bayes-risk expansions and Sanov information asymptotics, showing how $\log n$-order truncations arise naturally when risk, rather than pure exponents, is the evaluation criterion; (iii) we provide detailed applications to location testing under Laplace families, shape testing via Bayes factors, and connections to Fisher information geometry. The organizing principle throughout is that sample size enters Bayes-optimal GOF cutoffs through the MDP scale, unifying KS-based and Sanov-based perspectives under a single risk criterion.

This paper concerns the question of how AI systems encode semantic structure into the geometric structure of their representation spaces. The motivating observation is that the natural geometry of these representation spaces should reflect the way models use representations to produce behavior. We focus on the important special case of representations that define softmax distributions. In this case, we argue that the natural geometry is information geometry. Our focus is on the role of information geometry on semantic encoding and the linear representation hypothesis. As an illustrative application, we develop "dual steering", a method for robustly steering representations to exhibit a particular concept using linear probes. We prove that dual steering optimally modifies the target concept while minimizing changes to off-target concepts. Empirically, we find that dual steering enhances the controllability and stability of concept manipulation.

The generalized Pareto distribution (GPD) is a fundamental model for analyzing the tail behavior of a distribution. In particular, the shape parameter of the GPD characterizes the extremal properties of the distribution. As described in this paper, we propose a method for grouping shape parameters in the GPD for clustered data via graph fused lasso. The proposed method simultaneously estimates the model parameters and identifies which clusters can be grouped together. We establish the asymptotic theory of the proposed estimator and demonstrate that its variance is lower than that of the cluster-wise estimator. This variance reduction not only enhances estimation stability but also provides a principled basis for identifying homogeneity and heterogeneity among clusters in terms of their tail behavior. We assess the performance of the proposed estimator through Monte Carlo simulations. As an illustrative example, our method is applied to rainfall data from 996 clustered sites across Japan.

Longitudinal biomarkers are frequently collected in clinical studies due to their strong association with time-to-event outcomes. While considerable progress has been made in methods for jointly modeling longitudinal and survival data, comparatively little attention has been paid to statistical design considerations, particularly sample size and power calculations, in genetic studies. Yet, appropriate sample size estimation is essential for ensuring adequate power and valid inference. Genetic variants may influence event risk through both direct effects and indirect effects mediated by longitudinal biomarkers. In this paper, we derive a closed-form sample size formula for testing the overall effect of a single nucleotide polymorphism within a joint modeling framework. Simulation studies demonstrate that the proposed formula yields accurate and robust performance in finite samples. We illustrate the practical utility of our method using data from the Diabetes Control and Complications Trial.

Matrix approximation methods have successfully produced efficient, low-complexity approximate transforms for the discrete cosine transforms and the discrete Fourier transforms. For the DFT case, literature archives approximations operating at small power-of-two blocklenghts, such as \{8, 16, 32\}, or at large blocklengths, such as 1024, which are obtained by means of the Cooley-Tukey-based approximation relying on the small-blocklength approximate transforms. Cooley-Tukey-based approximations inherit the intermediate multiplications by twiddled factors which are usually not approximated; otherwise the effected error propagation would prevent the overall good performance of the approximation. In this context, the prime factor algorithm can furnish the necessary framework for deriving fully multiplierless DFT approximations. We introduced an approximation method based on small prime-sized DFT approximations which entirely eliminates intermediate multiplication steps and prevents internal error propagation. To demonstrate the proposed method, we design a fully multiplierless 1023-point DFT approximation based on 3-, 11- and 31-point DFT approximations. The performance evaluation according to popular metrics showed that the proposed approximations not only presented a significantly lower arithmetic complexity but also resulted in smaller approximation error measurements when compared to competing methods.

We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.

We consider the $\ell_p$-minimisation, which consists of finding the vector $x\in\mathbb{R}^N$ which minimises $\|x\|_p$ subject to the linear constraint $y=Ax$, where $y\in\mathbb{R}^m$ is given and $A$ is a $m\times N$ random matrix with i.i.d. sub-Gaussian centred entries ($m<N$). This can be viewed as the zero temperature version of a statistical mechanics problem, in which one introduces a suitable Gibbs measure on $\mathbb{R}^N$. To such a Gibbs measure there are associated belief propagation equations. We prove in the easiest case $p=2$ that the means of the distributions obtained by the belief propagation iteration satisfy asymptotically the approximate message passing equations.

Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.

Super-resolution is widely used in medical imaging to enhance low-quality data, reducing scan time and improving abnormality detection. Conventional super-resolution approaches typically rely on paired datasets of downsampled and original high resolution images, training models to reconstruct high resolution images from their artificially degraded counterparts. However, in real-world clinical settings, low resolution data often arise from acquisition mechanisms that differ significantly from simple downsampling. As a result, these inputs may lie outside the domain of the training data, leading to poor model generalization due to domain shift. To address this limitation, we propose a distributional deep learning framework that improves model robustness and domain generalization. We develop this approch for enhancing the resolution of 4D Flow MRI (4DF). This is a novel imaging modality that captures hemodynamic flow velocity and clinically relevant metrics such as vessel wall stress. These metrics are critical for assessing aneurysm rupture risk. Our model is initially trained on high resolution computational fluid dynamics (CFD) simulations and their downsampled counterparts. It is then fine-tuned on a small, harmonized dataset of paired 4D Flow MRI and CFD samples. We derive the theoretical properties of our distributional estimators and demonstrate that our framework significantly outperforms traditional deep learning approaches through real data applications. This highlights the effectiveness of distributional learning in addressing domain shift and improving super-resolution performance in clinically realistic scenarios.

Bayesian statistics is an integral part of contemporary applied science. bayesics provides a single framework, unified in syntax and output, for performing the most commonly used statistical procedures, ranging from one- and two-sample inference to general mediation analysis. bayesics leans hard away from the requirement that users be familiar with sampling algorithms by using closed-form solutions whenever possible, and automatically selecting the number of posterior samples required for accurate inference when such solutions are not possible. bayesics} focuses on providing key inferential quantities: point estimates, credible intervals, probability of direction, region of practical equivalance (ROPE), and, when applicable, Bayes factors. While algorithmic assessment is not required in bayesics, model assessment is still critical; towards that, bayesics provides diagnostic plots for parametric inference, including Bayesian p-values. Finally, bayesics provides extensions to models implemented in alternative R packages and, in the case of mediation analysis, correction to existing implementations.

We theoretically justify the recent empirical finding of [Teh et al., 2025] that a transformer pretrained on synthetically generated data achieves strong performance on empirical Bayes (EB) problems. We take an indirect approach to this question: rather than analyzing the model architecture or training dynamics, we ask why a pretrained Bayes estimator, trained under a prespecified training distribution, can adapt to arbitrary test distributions. Focusing on Poisson EB problems, we identify the existence of universal priors such that training under these priors yields a near-optimal regret bound of $\widetilde{O}(\frac{1}{n})$ uniformly over all test distributions. Our analysis leverages the classical phenomenon of posterior contraction in Bayesian statistics, showing that the pretrained transformer adapts to unknown test distributions precisely through posterior contraction. This perspective also explains the phenomenon of length generalization, in which the test sequence length exceeds the training length, as the model performs Bayesian inference using a generalized posterior.

Knowledge of the protection afforded by vaccines might, in some circumstances, modify a vaccinated individual's behaviour, potentially increasing exposure to pathogens and hindering effectiveness. Although vaccine studies typically do not explicitly account for this possibility in their analyses, we argue that natural direct effects might represent appropriate causal estimands when an objective is to quantify the effect of vaccination on disease while blocking its influence on behaviour. There are, however, complications of a practical nature for the estimation of natural direct effects in this context. Here, we discuss some of these issues, including exposure-outcome and mediator-outcome confounding by healthcare seeking behaviour, and possible approaches to facilitate estimates of these effects. This work highlights the importance of data collection on behaviour, of assessing whether vaccination induces riskier behaviour, and of understanding the potential effects of interventions on vaccination that could turn off vaccine's influence on behaviour.