Papers

Score-based generative models (SGMs) have achieved remarkable empirical success, motivating their application to a broad range of data distributions. However, extending them to heavy-tailed targets remains a largely open problem. Although dedicated models for heavy-tailed distributions have been proposed, their generative fidelity remains unclear and they lack solid theoretical foundations, leaving important questions open in this regime. In this paper, we address this gap through two theoretical contributions. First, we show that combining early stopping with a suitable initialization is sufficient to extend the diffusion framework to any target distribution; in particular, we establish the well-posedness of the backward process and prove convergence of the approximated diffusion in KL divergence. Second, we derive novel theoretical guarantees for generation with normalizing flows, obtaining convergence results that hold under mild conditions on the flow family and without any assumption on the tail behavior of the target distribution. Building on these results, we propose a unified generative framework for heavy-tailed distributions: a normalizing flow is first trained to capture the tail behavior and is then used as an initialization prior for an SGM, which refines the samples by recovering fine-grained structural details. This design leverages the complementary strengths of the two model classes within a theoretically principled pipeline, overcoming the limitations of existing approaches.

A simple integral representation involving no derivatives or continuity assumptions is given for proper single-event scoring rules.

Network meta-analysis (NMA) is widely used to compare multiple interventions simultaneously by synthesizing direct and indirect evidence. The general fixed or random effects contrast-based NMA model can be applied to different outcomes and data structures by opting for either an arm-based or contrast-based likelihood depending on the data available. Depending on the outcome and link-function, we estimate either collapsible or non-collapsible effect measures. Using an illustrative example involving binary outcomes and the non-collapsible odds ratio, we demonstrate that the standard NMA model produces estimates for non-collapsible effect measures that are biased toward the null when studies in the evidence base enroll heterogeneous populations (mixtures of distinct risk groups) that vary across studies. Importantly, this also holds when there are no differences in effect-modifiers across studies; the standard assumption of a common treatment effect when there are no differences in the distribution of effect-modifiers across studies is not appropriate when studies have different baseline risks. As a potential solution, we propose a “bookend” approach that explicitly models mixed-population studies as weighted combinations of two homogeneous subpopulations identified from studies with extreme baseline risks and provide guidance for practitioners to determine if bias due to non-collapsibility may be a concern.

Accurate power and sample size (PSS) calculations are essential for designing studies that use quasi-likelihood (QL) models, which extend generalized linear models (GLMs) to settings where the full distribution of the outcome is not specified. Traditional PSS approaches often rely on restrictive distributional assumptions, limiting their applicability when responses have non-standard distributions, variance functions are misspecified, or when predictors exhibit complex dependence structures. Building on recent advances in effect size measures for PSS - specifically, 2 Standard Deviations in the Linear Predictor (2SLiP) and Pseudo-Partial $R^2$ (P2R2) - developed with interpretability in mind, this paper extends and evaluates these effect size measures in the QL framework, keying in particular on their utility in PSS. We assess their empirical performance for the Wald test and then extend to the score test through extensive simulations across diverse outcome types, link functions, and variance structures. To illustrate practical utility, we applied these effect size measures to survey data on frontline health care workers from \citet{cahill2022occupational} to quantify the association between perceived personal protective equipment adequacy and mental health outcomes during the COVID-19 pandemic, adjusting for covariates. Our findings demonstrate that both 2SLiP and P2R2 provide robust and interpretable alternatives to traditional methods, maintaining accuracy with minimal distributional assumptions and enhancing the flexibility of PSS for realistic study designs.

We study computationally and statistically efficient reinforcement learning under the linear $Q^{\pi}$ realizability assumption, where any policy's $Q$-function is linear in a given state-action feature representation. Prior methods in this setting are either computationally intractable, or require (local) access to a simulator. In this paper, we propose a computationally efficient online RL algorithm, named Frozen Policy Iteration, under the linear $Q^{\pi}$ realizability setting that works for Markov Decision Processes (MDPs) with stochastic initial states, stochastic rewards and deterministic transitions. Our algorithm achieves a regret bound of $\widetilde{O}(\sqrt{d^2H^6T})$, where $d$ is the dimensionality of the feature space, $H$ is the horizon length, and $T$ is the total number of episodes. Our regret bound is optimal for linear (contextual) bandits which is a special case of our setting with $H = 1$. Existing policy iteration algorithms under the same setting heavily rely on repeatedly sampling the same state by access to the simulator, which is not implementable in the online setting with stochastic initial states studied in this paper. In contrast, our new algorithm circumvents this limitation by strategically using only high-confidence part of the trajectory data and freezing the policy for well-explored states, which ensures that all data used by our algorithm remains effectively on-policy during the whole course of learning. We further demonstrate the versatility of our approach by extending it to the Uniform-PAC setting and to function classes with bounded eluder dimension.

Two strategies are explored for robustifying classical denoising procedures for the Gaussian sequence model. First, the Hodges and Lehmann (1952) restricted Bayes approach is used to reduce sensitivity to the specification of the initial prior distribution. Second, alternatives to the Gaussian noise assumption are explored. In both cases proposals of Huber (1964) and Mallows (1978) play a crucial role.

For an exchangeable Bernoulli sequence with de Finetti mixing measure Pi, the k-step predictive probability P(X_{n+1}=...=X_{n+k}=0 | F_n) equals the posterior expectation E[(1-theta)^k | F_n]. By binomial expansion, this depends on all posterior moments up to order k. We show that the first moment alone is not sufficient to uniquely identify these quantities: for k >= 2, the mapping from posterior mean to k-step predictive is set-valued. The martingale posterior framework of Fong, Holmes, and Walker (which constrains only the first conditional moment of the terminal value) does not, in general, uniquely identify multi-step predictive distributions. Under any strictly proper scoring rule, the plug-in predictive is strictly dominated by the Bayes predictive whenever the posterior is non-degenerate. A closure theorem establishes that a martingale posterior determines all k-step predictives if and only if the conditional law of the terminal value is uniquely specified. Hill's A_{(n)} rule under the Jeffreys Beta(1/2,1/2) prior is a positive example. The discrepancy is O(Var(theta | F_n)) and vanishes as the posterior concentrates. These results clarify the structural requirements for predictive completeness under exchangeability.

We propose a retrodictive forecasting paradigm for time series: instead of predicting the future from the past, we identify the future that best explains the observed present via inverse MAP optimization over a Conditional Variational Autoencoder (CVAE). This conditioning is a statistical modeling choice for Bayesian inversion; it does not assert that future events cause past observations. The approach is theoretically grounded in an information-theoretic arrow-of-time measure: the symmetrized Kullback-Leibler divergence between forward and time-reversed trajectory ensembles provides both the conceptual rationale and an operational GO/NO-GO diagnostic for applicability. We implement the paradigm as MAP inference over an inverse CVAE with a learned RealNVP normalizing-flow prior and evaluate it on six time series cases: four synthetic processes with controlled temporal asymmetry and two ERA5 reanalysis datasets (wind speed and solar irradiance). The work makes four contributions: (i) a formal retrodictive inference formulation; (ii) an inverse CVAE architecture; (iii) a model-free irreversibility diagnostic; and (iv) a falsifiable validation protocol with four pre-specified predictions. All pre-specified predictions are empirically supported: the diagnostic correctly classifies all six cases; the learned flow prior improves over an isotropic Gaussian baseline on GO cases; the inverse MAP yields no spurious advantage on time-reversible dynamics; and on irreversible GO cases, it achieves competitive or superior RMSE relative to forward baselines, with a statistically significant 17.7% reduction over a forward MLP on ERA5 solar irradiance. These results provide a structured proof-of-concept that retrodictive forecasting can constitute a viable alternative to conventional forward prediction when statistical time-irreversibility is present and exploitable.

This paper is a follow-up to Maruyama and Strawderman (2006, Journal of Statistical Planning and Inference), which identified a new class of generalized Bayes estimators with a particularly simple form for estimating a normal variance under entropy loss. Although their previous work established the Bayesianity of these estimators, it did not provide a closed-form result for their minimaxity. In this paper, we revisit the problem and establish a definitive closed-form minimaxity result for this class of simple Bayes estimators.

Generative foundation models are increasingly scaled in both width and depth, posing significant challenges for stable feature learning and reliable hyperparameter (HP) transfer across model sizes. While maximal update parameterization ($\mu$P) has provided a principled solution to both problems for width scaling, existing extensions to the joint width-depth scaling regime remain fragmented, architecture- and optimizer-specific, and often rely on technically involved theories. In this work, we develop a simple and unified spectral framework for $\mu$P under joint width-depth scaling. For deep residual networks whose residual blocks contain $k$ transformations, the framework specifies how the norms of weights and their per-step updates should scale with width and depth. It reveals a fundamental transition from $k=1$ to $k\geq 2$, unifying previously disparate $\mu$P formulations and identifying the $k\geq 2$ case as more appropriate for practical architectures with multi-transformation branches such as Transformers. Building on this framework, we derive a general recipe for implementing $\mu$P across a broad class of optimizers by mapping spectral constraints to concrete HP parameterizations, recovering existing results and extending them to additional optimizers. Finally, experiments on GPT-2 style language models show that the $\mu$P formulation derived from the $k\geq 2$ case achieves stable feature learning and robust HP transfer under width-depth scaling, whereas standard parameterization and $\mu$P in the $k=1$ case often fail to do so. These results support the practical effectiveness of the proposed spectral framework.

Tourism demand forecasting is methodologically mature, but it typically treats accommodation supply as fixed or exogenous. In platform-mediated short-term rentals, supply is elastic, decision-driven, and co-evolves with demand through pricing, information design, and interventions. I reframe the core issue as endogenous stock-out censoring: realized booked nights satisfy B_{k,t} <= min(D_{k,t}, S_{k,t}), so booking models that ignore supply learn a regime-specific ceiling and become fragile under policy changes and supply shocks. This narrated review synthesizes work from tourism forecasting, revenue management, two-sided market economics, and Bayesian time-series methods; develops a three-part coupling framework (behavioral, informational, intervention); and illustrates the identification failure with a toy simulation. I conclude with a focused research agenda for jointly forecasting supply, demand, and their compositions.

There is recent interest in estimating the false discovery rate (FDR) with published p-values. However, there is little formal research that addresses the manner and extent to which the presumed selection, or publication, bias model impacts the bias and variance of FDR estimators. This manuscript provides general and closed-form expressions for the bias and variance of an established FDR estimator when the publication bias model (p<0.05) may or may not be correct. Expressions reveal that FDR estimates could be conservative or liberal, depending on how well a $p<0.05$ publication rule approximates the true selection mechanism. Analysis of a well-studied large-scale replication project in psychology, where selection model parameters are estimable, suggests that bias expressions are accurate in practice. Another well-studied collection of p-values mined from medical journal abstracts is used to illustrate how provided closed-form expressions may facilitate a simple sensitivity analysis when the goal is FDR estimation using selected p-values with unknown selection mechanism.

Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy, incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion (FWI), which estimates Earth's subsurface properties by fitting seismic measurements subject to the wave equation, where ill-conditioning stems from noisy, band-limited, finite-aperture measurements and complex geological structures. A Bayesian framework describes the solution more comprehensively: instead of a single estimate, a posterior distribution of plausible solutions characterizes the non-uniqueness and can be sampled to quantify uncertainty. However, no clear procedure exists for translating hard physical constraints, such as the wave equation, into priors amenable to existing sampling techniques. We address this by sampling the posterior in the dual space via an augmented Lagrangian formulation, which converts hard constraints into penalties suited to sampling algorithms while enforcing them progressively through multiplier updates, so they are satisfied in the limit. We integrate the alternating direction method of multipliers (ADMM) with Stein variational gradient descent (SVGD), a particle-based sampler: the constraint is relaxed at each iteration and the multiplier updates progressively enforce its satisfaction. This enables posterior sampling under hard constraints while inheriting the favorable conditioning of dual-space solvers, where partial constraint relaxation permits productive updates even when the current model is far from the true solution. We validate the method on a stylized Rosenbrock conditional inference problem and on frequency-domain FWI for a Gaussian anomaly model and the Marmousi II benchmark, demonstrating physically consistent uncertainty estimates and posterior contraction with increasing data coverage.

In this paper, we generate conceptual engineering designs of electric vertical take-off and landing (eVTOL) aircraft. We follow the paradigm of simulation-based inference (SBI), whereby we look to learn a posterior distribution over the full eVTOL design space. To learn this distribution, we sample over discrete aircraft configurations (topologies) and their corresponding set of continuous parameters. Therefore, we introduce a hierarchical probabilistic model consisting of two diffusion models. The first model leverages recent work on Riemannian Diffusion Language Modeling (RDLM) and Unified World Models (UWMs) to enable us to sample topologies from a discrete and continuous space. For the second model we introduce a masked diffusion approach to sample the corresponding parameters conditioned on the topology. Our approach rediscovers known trends and governing physical laws in aircraft design, while significantly accelerating design generation.

Inferring biophysical parameters and hidden state variables from partial and noisy observations is a fundamental challenge in computational neuroscience. This problem is particularly difficult for fast - slow spiking and bursting models, where strong nonlinearities, multiscale dynamics, and limited observational data often lead to severe sensitivity to initial parameter guesses and convergence failure in the methods replying on the traditional numerical forward solvers. In this work, we developed a physics-informed neural network (PINN) framework for the joint reconstruction of unobserved state variables and the estimation of unknown biophysical parameters in neuronal models. We demonstrate the effectiveness of the method on biophysical neuron models, including the Morris-Lecar model across multiple spiking and bursting regimes and a respiratory model neuron. The method requires only partial voltage observations over short observation windows and remains robust even when initialized with non-informative parameter guesses. These results suggest that PINN can deliver robust and accurate parameter inference and state reconstruction, providing a promising alternative for inverse problems in multiscale neuronal dynamics, where traditional techniques often struggle.

Surveys are critical inputs for research and policy, yet, enumerating a sampling frame is logistically infeasible or financially nonviable in many circumstances, such as during pandemics, natural disasters, or armed conflict. Respondent Driven Sampling (RDS) does not require a sampling frame, yet non-random peer recruitment often introduces substantial bias, particularly under high homophily. We introduce and evaluate Randomized Recruitment Driven Sampling (RRDS), a cellphone-based adaptation of RDS that incorporates researcher-controlled randomization into each recruitment wave. While standard RDS is necessary for stigmatized groups where network transparency is infeasible, RRDS is designed for low-stigma populations that become difficult to access due to logistical barriers. In these contexts, RRDS enforces the random recruitment assumption that traditional RDS relies upon but rarely achieves. Through simulation and an experiment surveying Bangladeshi garment workers during the COVID-19 pandemic, we demonstrate that RRDS produces less biased estimates and improved confidence interval coverage compared to traditional RDS. RRDS offers a scalable, remote-compatible alternative for studying low-stigma groups in challenging contexts where large-scale probability sampling is unsafe or infeasible.

Bayesian inference in generalized linear models requires a prior on the coefficient vector $\beta$. Practitioners naturally reason about response probabilities at specific covariate values, not about abstract log-odds parameters. We develop synthetic priors: informative Bayesian priors for GLMs grounded in Good's device of imaginary observations – the principle that every conjugate prior is equivalent to a likelihood on pseudo-data from the same exponential family. The conditional means prior of Bedrick (1996) elicits independent Beta priors on the conditional mean response at $p$ expert-chosen design points; the induced prior on $\beta$ is a product of binomial likelihoods at synthetic data points. Combined with Pólya-Gamma data augmentation \citep{polson2013}, the posterior admits an exact conjugate Gibbs sampler – no tuning, no Metropolis step – by treating the augmented dataset as a standard logistic regression. We show that ridge regression and catalytic priors \citep{huang2020} are instances of Good's device, and identify prediction-powered inference \citep{angelopoulos2023ppi} as a structural analogue in the frequentist setting – all three mediate a variance-bias tradeoff through a single informativeness parameter. We illustrate the approach on two benchmark problems: the Challenger O-ring data \citep{dalal1989}, where the BCJ prior provides a more moderate posterior predictive at the 31{\deg}F launch temperature; and a Phase~II atopic dermatitis dose-finding trial ($n = 300$), where the synthetic prior narrows 95% credible intervals by 3-6% and raises decision probabilities by up to 2 percentage points relative to a flat prior.

In observational studies, causal inference becomes difficult when confounders are missing-not-at-random (MNAR), particularly where the missingness depends on the confounder's own unreported value (self-masking). Existing methods for handling MNAR confounders often rely on strong, unverifiable assumptions, leading to biased estimates. We propose a simple approach with Stratified Delta-Imputed Propensity Estimator (SDIPE) in the presence of self-masking confounders. SDIPE first stratifies data into observed and missing groups, imputes missing confounders via delta-adjusted multiple imputation. Then, within each group, average-treatment-effects (ATEs) are estimated by stabilized-inverse-probability-weights. The final ATE is obtained by combining the subgroup-specific estimates, weighted by respective proportions in the sample. Simulation study shows that SDIPE achieves low bias and near-nominal coverage (94-96%) across varying missingness, sample sizes, and treatment prevalence. In contrast, conventional sensitivity-based multiple imputation exhibits substantial bias and poor coverage (18-89%). Additionally, SDIPE is robust to the choice of the delta parameter. Applied to NHANES-2017-2018, SDIPE estimates that married individuals have a 1.19-point lower depression score than unmarried individuals (95% CI: -1.76, -0.64), adjusting for MNAR income data. SDIPE provides a practical and robust approach for causal inference with self-masking MNAR confounders, offering improved performance over existing methods without requiring restrictive assumptions about the missingness mechanism.

This paper investigates numerical solution methods for the Schatten-$p$ quasi-norm regularized problem with $p \in [0,1]$, which has been widely studied for finding low-rank solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. We propose a dynamic proximal gradient algorithm that, through the use of the Cayley transformation, avoids computationally expensive singular value decompositions at each iteration, thereby significantly reducing the computational complexity. The algorithm incorporates two step size selection strategies: an adaptive backtracking search and an explicit step size rule. We establish the sublinear convergence of the proposed algorithm for all $p \in [0,1]$ within the framework of the Kurdyka-Lojasiewicz property. Notably, under mild assumptions, we show that the generated sequence converges to a stationary point of the objective function of the problem. For the special case when $p=1$, the linear convergence is further proved under the strict complementarity-type regularity condition commonly used in the linear convergence analysis of the forward-backward splitting algorithms. Preliminary numerical results validate the superior computational efficiency of the proposed algorithm.

With the increasing availability of data objects in the form of probability distributions, there is a growing need for statistical methods tailored to distributional data. Distance measures, especially the pairwise distance matrix between data objects, provide the foundation for a wide range of modern data analysis methods, such as clustering, multidimensional scaling, and distance-based regression, among others. The Wasserstein distance is commonly used with distributional data due to its compelling optimal transport property. However, while the Wasserstein distance can be efficiently computed for univariate distributions, its application to multivariate distributions is limited due to high computational costs. To address these scalability issues, we introduce the Nonparanormal Transport (NPT) metric, a closed-form distance based on the flexible nonparanormal distribution family for modeling skewed and non-Gaussian multivariate data. Simulation studies demonstrate that NPT maintains a high level of agreement with the Wasserstein distance, while being at least 1000 times faster than its efficient variants when computing a 100-distribution pairwise distance matrix in both 2 and 5 dimensions. We illustrate the utility of NPT through a multidimensional scaling analysis of bivariate oxygen desaturation distributions of 723 individuals with sleep apnea in the Sleep Heart Health Study.

Upon taking office in late 2012, Chinese President Xi Jinping launched one of the most intensive anticorruption campaigns in the history of the People's Republic of China. Prior to the campaign, China's land market suffered from corruption, particularly surrounding sale method selection (auction versus listing). Listing is a two-stage sale mechanism that prior research has identified as more susceptible to corruption, leading to lower prices. This paper examines the campaign's impact on land allocation, focusing on whether corruption influences the choice of sale method and, in turn, land sale prices. This paper is the first to utilize Blackwell and Yamauchi (2021, 2024)'s marginal structural model with fixed effects in the inverse probability of treatment weighting model; absorbing time-invariant unobserved confounding and utilizing a set of time-varying covariates as controls, this model can estimate causal effects in the land sale case. I find that indictments in a prefecture cause a statistically significant drop in the probability that land is sold via listing$\unicode{x2014}$an effect that is further compounded when indictments occur in consecutive months. Sensitivity analyses indicate that any violations of the identification assumptions would bias estimates towards zero, confirming the negative effect. A second marginal structural model shows that both mean and median land sale prices increase in the presence of indictments. Together, these results suggest that the anticorruption campaign not only deterred actual corrupt allocation practices, but also impacted the discretionary use of listings.

We propose the CliPS procedure when fitting Bayesian mixture models in the context of model-based clustering to identify the cluster distributions while simultaneously assessing the suitability of a cluster solution and validating the cluster structure. The procedure relies on the point process representation of a mixture model and is based on the assumption that a suitable cluster solution requires the clusters to be distinguishable with respect to a low-dimensional functional of the component-specific parameters of the mixture. CliPS maps the component-specific MCMC draws to the point process representation and identifies clusters there, exploiting that, while data distributions usually overlap, the posterior of these functionals are more and more separated for increasing sample size. We outline the procedure and illustrate its use on several model-based clustering examples.

Linear regression estimators are known to be sensitive to outliers, and one alternative to obtain a robust and efficient estimator of the regression parameter is to model the error with Student's $t$ distribution. In this article, we compare estimators of the degrees of freedom parameter in the $t$ distribution using frequentist and Bayesian methods, and then study properties of the corresponding estimated regression coefficient. We also include the comparison with some recommended approaches in the literature, including fixing the degrees of freedom and robust regression using the Huber loss. Our extensive simulations on both synthetic and real data demonstrate that estimating the degrees of freedom via the adjusted profile log-likelihood approach yields regression coefficient estimators with high accuracy, performing comparably to the maximum likelihood estimators where the degrees of freedom are fixed at their true values. These findings provide a detailed synthesis of $t$-based robust regression and underscore a key insight: the proper calibration of the degrees of freedom is as crucial as the choice of the robust distribution itself for achieving optimal performance. The R package that implements our method is available at https://github.com/amanda-ng518/RobustTRegression.

While many statistical procedures rely on a fixed sample size, sequential methods allow a decision-maker to adapt the sample size to achieve a given precision. In this way, sequential tests reduce the average number of observations required to achieve a given power of the test – but by how much? To address this question, we focus on the scenario of testing the unknown drift of a Brownian motion, comparing the Wald sequential probability ratio test with tests that use a pre-determined fixed sample size. We provide precise bounds on the average reduction in sample size needed to achieve a desired precision. Specifically, we demonstrate that for symmetric error bounds, the sequential test reduces the average sample size by at least 36% and by at most 75%. Moreover, the reduction in sample size increases monotonically with the power of the test, meaning that the relative advantage of using a sequential test over a fixed sample size test grows as higher power is required. We also study the relative efficiency in the case with asymmetric error bounds, and we provide a lower bound in terms of the symmetric case.

We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed non-equal volume partitions yield stratified sampling point sets with lower expected star discrepancy than classical jittered sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$, where $Y$ and $Z$ represent jittered sampling and our non-equal volume partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our non-equal volume partition models, which improve upon existing bounds for jittered sampling. Our results provide a theoretical foundation for using non-equal volume partitions in high-dimensional numerical integration.

In healthcare, predictive models increasingly inform patient-level decisions, yet little attention is paid to the variability in individual risk estimates and its impact on treatment decisions. For overparameterized models, now standard in machine learning, a substantial source of variability often goes undetected. Even when the data and model architecture are held fixed, randomness introduced by optimization and initialization can lead to materially different risk estimates for the same patient. This problem is largely obscured by standard evaluation practices, which rely on aggregate performance metrics (e.g., log-loss, accuracy) that are agnostic to individual-level stability. As a result, models with indistinguishable aggregate performance can nonetheless exhibit substantial procedural arbitrariness, which can undermine clinical trust. We propose an evaluation framework that quantifies individual-level prediction instability by using two complementary diagnostics: empirical prediction interval width (ePIW), which captures variability in continuous risk estimates, and empirical decision flip rate (eDFR), which measures instability in threshold-based clinical decisions. We apply these diagnostics to simulated data and GUSTO-I clinical dataset. Across observed settings, we find that for flexible machine-learning models, randomness arising solely from optimization and initialization can induce individual-level variability comparable to that produced by resampling the entire training dataset. Neural networks exhibit substantially greater instability in individual risk predictions compared to logistic regression models. Risk estimate instability near clinically relevant decision thresholds can alter treatment recommendations. These findings that stability diagnostics should be incorporated into routine model validation for assessing clinical reliability.

In this article, bipartite ranking, a statistical learning problem involved in many applications and widely studied in the passive context, is approached in a much more general active setting than the discrete one previously considered in the literature. While the latter assumes that the conditional distribution is piece wise constant, the framework we develop permits in contrast to deal with continuous conditional distributions, provided that they fulfill a Hölder smoothness constraint. We first show that a naive approach based on discretisation at a uniform level, fixed a priori and consisting in applying next the active strategy designed for the discrete setting generally fails. Instead, we propose a novel algorithm, referred to as smooth-rank and designed for the continuous setting, which aims to minimise the distance between the ROC curve of the estimated ranking rule and the optimal one w.r.t. the $\sup$ norm. We show that, for a fixed confidence level $\epsilon>0$ and probability $\delta\in (0,1)$, smooth-rank is PAC$(\epsilon,\delta)$. In addition, we provide a problem dependent upper bound on the expected sampling time of smooth-rank and establish a problem dependent lower bound on the expected sampling time of any PAC$(\epsilon,\delta)$ algorithm. Beyond the theoretical analysis carried out, numerical results are presented, providing solid empirical evidence of the performance of the algorithm proposed, which compares favorably with alternative approaches.

Background: The E-value has become widely used for assessing robustness to unmeasured confounding in observational studies, but the original framework was developed for single time-point exposure-outcome settings. This study extends the E-value methodology to longitudinal set up with time-varying treatments and confounders, where treatment-confounder feedback occurs. Methods: A combined bias factor accounting for unmeasured confounding at multiple time points was extended, with three reporting scenarios presented: equal bias distribution across time points, confounding at a single time point, and a general case visualizing all possible confounder strength combinations. Results: In simulations with an observed risk ratio of 1.73, unmeasured confounders with 1.96-fold associations at each time point could nullify the effect under equal distribution-substantially lower than the single time-point E-value of 2.85. Re-analysis of a published insulin resistance and cardiovascular disease study yielded similar patterns, with time-varying E-values of 1.63 at each time point compared to the originally reported 2.09. Conclusions: Studies more like longitudinal set up may be more vulnerable to unmeasured confounding than single time-point E-values suggest. This extension provides accessible tools for transparent sensitivity analysis in time-varying settings while preserving the simplicity and minimal assumptions that make E-values widely applicable.

Estimation of the population total of a variable can be improved by calibration on a set of auxiliary variables. It is difficult to establish that such a set of variables is sufficient, that estimation could not be improved by calibration on any further variables. We address this issue by finding an upper bound for the change of the calibration estimate of the population total of a variable when the auxiliary information is supplemented by another variable for which the population total is known. This upper bound can be interpreted as a measure of sensitivity of the estimate to unavailable auxiliary information and considered as a factor in deciding whether to seek further data sources that would be included in calibration.

In probabilistic classification, calibration error (CE) measures the average divergence of predicted probabilities $f(X)$ from $\mathbb{P}(Y|f(X))$, the true class distribution for that predicted probability. While being a useful diagnostic tool, it is hard to estimate: popular binning-based estimators are often inconsistent and scale poorly beyond two classes. Recent work rewrites the CE as the excess risk of a model compared to the best recalibration of its own predictions, measured with a proper loss. However, this only works for Bregman-divergence-based calibration errors like the squared error, excluding the more popular $L_1$-distance-based CE. We show that using prediction-dependent proper scores can alleviate this restriction, allowing us to estimate CEs with general convex divergences, including $L_p$ distances with closed-form losses in the binary and multiclass settings. To estimate the excess risk, we introduce a more accurate recalibrator that fits a residual to temperature scaling with gradient boosting. The resulting variational estimator, V-ECE, needs no bins or clusters and lower-bounds the true calibration error in expectation. On a benchmark of semi-synthetic tasks built from real classifiers, with known true CE, V-ECE is among the most accurate binary estimators for every calibration error and significantly outperforms all multiclass estimators. Our results are accompanied by additional theory on $L_p$ CE, estimator bias, and over- or under-confidence estimation.