Papers

In this article, we address the problem of uncertainty quantification of state variables in the context of inverse problems. Inverse problems are associated with phenomena that can be represented through ordinary or partial differential equations, for which observations or data are available, but the values of the parameters that characterize the equations, the initial or boundary conditions are not known. Currently, the literature offers very few alternatives for analyze the propagation of uncertainty of state variables, which are limited to constructing pseudo-credible regions or quantifying their uncertainty at isolated points. We propose a methodology that combines tools of Bayesian inference, with Hamiltonian Monte Carlo sampling employed for efficient posterior exploration, and functional data analysis, specifically the Modified Band Depth method, to determine credible regions for the trajectories of the state variables throughout the time horizon of interest. We expose a methodology validation through a simulation study, which shows that our proposal captures a higher proportion of state variable trajectories than traditional pointwise analysis methods, our proposal generated credible regions that contained the true trajectory of the state variables 96.4% of the times, versus 80% that the regions of the pointwise method did. Furthermore, we demonstrate its application to a non-trivial model associated with a neurological phenomenon, for which the methodology effectively captures the time-dependent dynamics of the state variables.

Functional linear models are commonly used for analyzing functional data with scalar responses, particularly for modeling the conditional mean of the response given functional covariates. In this work, we extend the traditional scalar-on-function regression in two major aspects: 1. estimating the conditional distribution function instead of a particular characteristic such as the conditional mean; 2. considering an arbitrary operator without imposing functional linearity or any model assumption. We use a likelihood approach for the conditional hazard function and apply convolutional neural networks to approximate the effects of functional inputs and estimate the conditional distribution using deep operator networks. Through simulations and a real world data example, we show the desirable robustness of the proposed method in comparison with the mean regression neural networks, demonstrating that our approach achieves better conditional distribution estimation and interval prediction in complex data settings.

Many statistical models involve parameter-dependent normalizing constants that are computationally intractable, creating substantial obstacles to standard Bayesian inference. Although existing likelihood-based algorithms can often circumvent these constants, their uncertainty quantification may be poorly calibrated under model misspecification. To address these challenges, we propose SME-BETEL, a semiparametric Bayesian framework that combines score matching estimating equations with Bayesian exponentially tilted empirical likelihood. The resulting posterior avoids evaluation of normalizing constants and does not require learning-rate calibration. Building on this framework, we develop a new score matching criterion for mixed-domain data, extending SME-BETEL to models whose observations combine components from different sample spaces. This construction enables robust Bayesian inference for mixed-domain doubly-intractable models. We establish consistency and asymptotic normality of the score matching estimator, and prove a Bernstein-von Mises theorem for the SME-BETEL posterior. These results show that SME-BETEL credible sets are asymptotically calibrated to the sampling variability of the score matching estimator, yielding valid frequentist coverage under model misspecification. Simulation studies show that SME-BETEL remains competitive under correct specification and substantially improves uncertainty quantification under misspecification. An ozone-monitoring application demonstrates the practical utility of the mixed-domain construction for spatial preferential modeling.

In many inverse problems, the aim is to recover a probability distribution on a latent (or object state) space from indirect, noisy observations. When the observations can be modelled as noisy samples from the mixing latent distribution, the recovery problem is a deconvolution problem on the space of probability measures. A common strategy is to fix a finite set of candidate points and estimate a weight for each, turning the problem into a finite-dimensional concave maximum likelihood problem on the simplex. We study the combined effect of this discretization and of the noise in the context of cryo-electron microscopy (cryo-EM), where the candidate points are biomolecular conformations and the weights describe the relative frequency of each conformation. Our results pertain to both statistical and algorithmic aspects of the estimator. We analyze the weight-recovery problem in which the candidate states and their likelihoods are known. A nearby pair of candidates forces a near-null direction. More generally, the grid and the noise level impose a uniform lower bound on the achievable Kullback–Leibler divergence between observation densities, even with infinite data. The finite-grid estimator is asymptotically normal when its population target is in the interior of the simplex; at a boundary target, its limit is a cone-projected Gaussian. Finally, the exact proximal form of Expectation–Maximization leads to a global high-noise comparison between an early iterate and a KL-penalized likelihood, without a basin assumption or a linearization of the recursion. Tests on synthetic images of the Hsp90 molecule illustrate the theoretical findings and translate them into practical guidelines for interpreting reweighted ensembles.

We introduce a statistical framework for Gaussian fields indexed at arbitrary edge locations on general compact directed metric graphs. The construction is based on a stochastic differential equation with a first-order operator and conditions at the vertices. We characterise well-posedness and identify the covariance reproducing kernel Hilbert space. We also connect the proposed framework to earlier stream-network models, showing that these arise from the same system under particular boundary conditions, and introduce new boundary conditions that yield more physically realistic processes. The differential-equation representation enables computationally efficient inference and prediction. This makes the method applicable to large data sets without approximation. Applications to temperature modelling on river networks and traffic speeds on road networks illustrate the framework, including the computational efficiency and improved performance under physically informed vertex conditions.

Multinomial probit choice probabilities over n alternatives are Gaussian orthant integrals, computed by simulation for thirty years, one expensive integral per alternative. Inversion, which is to say determining item attractiveness consistent with a prescribed choice probability vector, is even more difficult and has been considered impractical for correlated contests when n is large. Yet here, for families lying within a grammar including factor, block and hierarchical covariance structures, we exhibit a calibration tested at n = 1,000,000 reproducing probabilities to very high accuracy, even in the extreme tail. We must return to much smaller problems for any performance comparison to be possible due to limitations of the prior art. The Geweke-Hajivassiliou-Keane simulator is the standard (and still appropriate for high rank) but is two hundred times slower already at n = 200, and its measured cost grows roughly as n^2.8 while ours is linear. Furthermore our approach applies to any continuous performance distributions within reason: the Thurstone-Mosteller model families thereby become a practical alternative to logit at modern scale.

Conditional independence tests (CITs) test for conditional dependence between two random objects $X$ and $Y$ given a third random object $Z$. Existing CITs have limited applicability to high-dimensional data, especially multimodal data like text. However, we show that such tests are of interest for large language model (LLM) outputs, where we test whether an output $X$ generated from a source text $Z$ carries information about an attribute $Y$ beyond $Z$ itself. For this purpose, we propose embedded CITs (eCITs), which embed $X$ and $Z$ and apply an existing CIT to the resulting representations and to $Y$. We show that, provided the embedding of $Z$ is sufficient, i.e. retains the information $Z$ carries about either $Y$ or the representation of $X$, the null hypothesis transfers from $X$ and $Z$ to their representations, so that a CIT valid for the embedded hypothesis is valid for the original one. We further give conditions for equivalence of the two hypotheses, and show that sufficiency weakens to mean sufficiency when the embedded test targets conditional mean independence. We propose a semi-synthetic simulation design to assess type I error (T1E) control and power of the eCITs for given embedding maps on a specific dataset and task, and use it to evaluate them on our application. Applying the eCITs to German Parliament speeches, we find for all combinations of embedding maps considered that the summaries of two LLMs contain information about the speaker's faction and gender beyond the speech they were generated from.

High-dimensional clustering is challenging when component distributions are both heavy-tailed and directionally asymmetric. We propose a deep skew-$t$ mixture model (DStMM), a hierarchical factor-analytic mixture based on the generalised-hyperbolic skew-$t$ normal mean–variance representation. A shared inverse-gamma mixing variable is propagated along each complete latent pathway, allowing heavy tails and directional asymmetry to be modelled jointly while preserving conditional Gaussianity. Each complete pathway therefore admits an exact GHST marginal representation. We formalise the reductions to symmetric deep $t$, Gaussian deep-mixture, and single-layer GHST factor-analytic models, discuss local non-identifiability and the implementation-level parameter-counting convention, and derive the conditional generalised inverse Gaussian law used for estimation. Estimation is carried out by a stochastic/Monte Carlo EM algorithm, with an explicit implementation-based parameter count for BIC architecture comparison. Simulation studies show that DStMM performs similarly to the symmetric robust model when skewness is absent but provides increasing gains as directional asymmetry becomes stronger, particularly under heavier tails; the same qualitative behaviour persists under smaller samples and unequal mixture proportions. Two real-data applications provide complementary evidence. On the UCI handwritten-digit benchmark, DStMM gives the strongest clustering performance under a common deep architecture, while on the Gas Sensor Array Drift data, DStMM improves on both deep Gaussian and deep $t$ alternatives and, under the implemented BIC criterion, selects a non-trivial second mixture layer. Together, these results support the value of propagating skewness and heavy-tail variation through a deep latent mixture while retaining an exact pathway-level likelihood.

Fixed-width sequential stopping rules terminate a stochastic simulation once an estimated confidence interval reaches a prescribed width. Classical fixed-width theory typically relies on a strongly consistent estimator of the asymptotic variance. This makes the normalized stopping time asymptotically deterministic, allowing fixed-sample-size limit theory to be transferred to the estimator at termination. This mechanism can fail when simulation output has infinite variance or long-range dependence. Although self-normalization and subsampling can yield asymptotically valid confidence intervals at a fixed sample size, the scaling process and the stopping time may retain nondegenerate randomness, so fixed-sample-size quantiles need not provide valid coverage at termination. In this paper, we develop a unified framework based on a joint functional limit theorem for the estimation process and a scaling process. We characterize the asymptotic behavior of both the stopping time and the self-normalized estimator evaluated at termination, thereby obtaining asymptotically valid sequential confidence intervals in classical finite-variance and infinite-variance settings. Moreover, we introduce a sequential subsampling procedure that consistently estimates the distribution relevant at the stopping time without directly estimating nuisance parameters in the limit distribution. The framework is verified for heavy-tailed moving-average processes, stochastic approximation, and an M/G/1 queue with heavy-tailed service times.

We introduce a closed-form, refit-free procedure for leave-one-cluster-out (LOCO) cross-validation in multilevel Gaussian Bayesian structural equation models (SEMs), together with predictive scoring of every nested submodel. Conditional independence of clusters given the parameters expresses the cluster-deleted posterior as a functional of the full posterior. The LOCO predictive density is then a harmonic mean of the cluster likelihood, computable from a single fit in INLAvaan, the integrated nested Laplace approximation package for Bayesian SEM. We evaluate the harmonic-mean expectation in closed form through a fully exponential Laplace approximation of the reciprocal cluster likelihood under a Gaussian posterior; candidate structural restrictions follow by Gaussian conditioning of the same Laplace summary. The resulting Taylor elpd (expected log predictive density) scores are fully deterministic, requiring neither refitting nor Monte Carlo sampling, so the variance pathology of the naive harmonic-mean estimator does not arise. A direct-sum decomposition of the compound-symmetric cluster covariance makes the cost independent of cluster size, orders of magnitude below brute-force refitting. We validate against brute-force refits and Markov chain Monte Carlo in simulation, and illustrate the procedure on school-safety climate in PISA 2022 and on 16 candidate structures linking personality and well-being in the MIDUS sibling sample.

Matrix-variate data with missing entries arise frequently in applications where observations are naturally organized as two-dimensional arrays. Although the matrix normal distribution provides a parsimonious model through its Kronecker covariance structure, standard EM estimation can be computationally expensive because arbitrary missingness patterns typically destroy this separability in the E-step. In this paper, we propose an efficient partial EM algorithm for matrix-variate normal data with missing entries. The proposed method updates the conditional mean and covariance of the missing component through coordinate-wise approximations, avoiding repeated inversion of pattern-specific covariance matrices and avoiding construction of the full vectorized covariance matrix. We further develop a specialized update for submatrix missingness, where the missing-block precision retains a Kronecker product structure, and the covariance update can be carried out independently in the row and column directions. Simulation studies show that the proposed methods substantially reduce computation time compared with exact EM while preserving nearly identical observed-data likelihood across a range of dimensions and missing proportions. A real-data application to hyperspectral image patches demonstrates that the proposed imputation strategy can be embedded within a matrix-variate mixture model for simultaneous imputation and clustering.

Many sequential distribution shift detectors update a growing reference set with incoming observations. After a change, this update contaminates the reference set with post-change observations and can weaken subsequent evidence. Keeping the calibration sample fixed mitigates this contamination, but repeated reuse induces dependence among fixed-reference ranks, so arguments based on independent conformal \(p\)-values do not apply. We derive the exact conditional null distribution of the next rank given the preceding ranks and use it to construct a predictive rank martingale (PRM). Thresholding a PRM gives distribution-free, finite-sample anytime marginal type I error control. To target specific departures, we apply a pre-specified feature to each rank, center the resulting payoff under the predictive null law, and use Online Newton Step (ONS) to adapt the bet. Order and dispersion features target directional and center-versus-tail changes, respectively. For any Lipschitz feature with nonzero induced contrast under the alternative, we establish a finite-window detection guarantee and show that the test is consistent as the initial calibration size increases. At a fixed calibration size, however, we derive a power ceiling for every distribution-free detection procedure. Across synthetic and real data, our PRM methods achieve better detection performance than conditional conformal test martingale (CCTM).

Evaluations that use a large language model (LLM) as a judge have begun to borrow reliability statistics from classical test theory and its extensions. We examine three such statistics that need one administration and no gold labels. None of them can isolate the judge, because one judge under one prompt supplies no variance component of its own. Claude Haiku 4.5 judged 210 constructed short answers against ten-element checklists. On the 180 with parsed verdicts, the Kuder-Richardson coefficient (KR-20) came out at 0.5223 on the judge's verdicts and 0.5231 on error-free gold verdicts. In simulation, bank design alone moves KR-20 from 0.01 to 0.68 at the judge's measured 4.72% error rate. The dependability index $\Phi(\lambda)$, a ratio of mean squared distances from the pass mark, sits 0.22 to 0.38 below the judge's accuracy against gold and returns 0.54 to 0.68 on error-free gold verdicts. Livingston-Lewis accuracy treats the rubric elements as a sample, and at a pass mark of five elements it credits error-free gold scores with 0.78, close to the judge's 0.81. A statement about the judge therefore needs gold labels or a varied scorer facet, and a reliability ratio needs the bank's spread beside it. One of the four closest judge-evaluation papers varies the prompt and still reads a reliability below 0.7 as a sign that a model cannot serve as a judge, although that reliability moves with the spread of the samples scored. We derive a decision table and four reporting lines from these two rules.

Bayesian synthesis, which generates synthetic data by sampling from the posterior predictive distribution, is a popular approach for privatizing sensitive personal data. However, how attribute disclosure risk is affected by feature dimensionality, the number of individuals in the original dataset, and the amount of released synthetic information remains poorly understood. We propose a mathematically tractable Bayesian synthesizer for multivariate normal data based on a conjugate normal-inverse-Wishart prior for the mean vector and covariance matrix. The conjugate structure yields closed-form posteriors and enables direct investigation of an adversary's ability to infer records under different data and release settings. We then demonstrate several intuitive properties of synthetic data generation through several simulations. Specifically, we show that disclosure risk decreases with the size of the original dataset, but increases with the dimensionality of the feature space and the amount of synthetic information released, whether through the release of larger synthetic datasets or multiple generator realizations. Finally, the proposed synthesizer was used to generate synthetic versions of a polygenic risk score dataset, with the synthetic data exhibiting distributional properties comparable to those of the original data.

Set-based genetic association tests provide a powerful framework for detecting genetic effects on complex traits by jointly analyzing multiple genetic variants. Although set-based methods have been developed for interval-censored survival outcomes, existing approaches primarily focus on a single survival phenotype and therefore do not fully use information from multiple correlated outcomes. In this paper, we develop two Weighted V Tests for Multivariate Interval-Censored Data (WV-M-IC), extending the weighted V-statistic framework (Wu et al., 2021) to the joint analysis of multiple correlated interval-censored survival outcomes. The performance of these methods is evaluated through simulation studies, showing that the proposed approaches can provide power gains compared with single-outcome analyses. We apply the proposed methods to the ZOE 2.0 study to investigate dental caries progression in children.

This paper develops a general convolutional neural network (CNN) framework for detecting heterogeneous event-driven dynamics in univariate time series windows. We show that the induced CNN class exactly represents classifiers based on range, maximum drawup, maximum drawdown and slope change, and uniformly approximates realised volatility and autoregressive explosiveness on compact domains. We further establish error bounds for representative rules in finite samples and an oracle inequality for learning across them. Simulations show that the proposed model can match or outperform classifiers based on individual statistics as the training sample grows. In an application to six daily energy price series, a hierarchical CNN distinguishes event windows and event families. Applied without retraining to observations withheld after 20 February 2026, the fitted model identifies predominantly geopolitical dynamics in several oil and refined product series around the outbreak of the 2026 Iran war, while distinguishing a contemporaneous natural gas spike associated with weather.

We propose a generalized Bayesian clustering with regression model for unaligned longitudinal binary outcomes, motivated by seizure diary data from the Human Epilepsy Project. Seizure diaries are sparse, irregularly observed, and vary enormously across patients. A single fully-specified generative model tends to be either misspecified or computationally inefficient. We address the challenge by two strategies. We set up a regression by way of clustering as model-based clustering using a mixture model. For the latter, we take a generalized Bayesian perspective which replaces the full likelihood with a loss-based update using a generalized likelihood. We combine a trajectory similarity loss and a regression loss, so that clustering is informed by both trajectory similarity and the prediction of outcomes The trajectory similarity loss is constructed by representing each trajectory as an (empirical) distribution of subsequences, called reads, and then is defined based on the sliced Wasserstein distance between these empirical distributions. This loss allows alignment-free comparison of sequences that are irregularly observed or temporally misaligned, and it scales quasi-linearly in trajectory length. The regression loss is the negative log-likelihood of a probit regression. A prior on the cluster-specific parameters is defined by way of a Dirichlet process prior on the mixing measure.

Gaussian graphical models (GGMs) describe the dependence structure among jointly Gaussian random variables. However, the most common parameterisation of GGMs, the precision matrix, describes both the dependence and scale of the variables. This has been shown to lead to model selection methods that depend on the scale of the variables, despite graphical models being scale invariant. Even after standardising data to have unit sample variances, entries of the precision matrix can be on different scales leading to poor model selection. This paper decomposes the precision matrix into marginal variances and interpretable scale-invariant parameters - the partial correlations and variance inflation factors. This decomposition gives new insights into the precision matrix entries and the dynamics of model selection methods such as penalised likelihoods. In particular, it explains the observed phenomenon that penalties on the precision matrix perform poorly at selecting hub variables and motivates the necessity of data standardisation. It also shows why methods based on partial correlations have better hub detection properties. The effect of penalisation of different quantities on the estimation of marginal variances is then investigated and an interesting simplification of the log-likelihood is shown when using maximum likelihood estimation of the marginal variances.

Psychometric questionnaires contain rich item-level information, yet it remains unclear whether different representation learning objectives recover the same latent organization. We investigated this question using 757 matched teacher-child pairs from the baseline assessment of the Cyprus ProW preschool trial. Behavioral structure was characterized from child SDQ, ASBI, and CBRS item responses using principal component analysis and clustering, yielding four behavioral phenotypes. A contrastive objective substantially improved teacher-child retrieval relative to PCA-based representations, increasing Top-1 accuracy from 0.13% to 7.27% and Top-10 accuracy from 1.98% to 56.14%. However, contrastive representations preserved behavioral phenotype structure less effectively than PCA-based representations. A multi-task objective jointly optimizing alignment and behavioral prediction partially restored behavioral organization but reduced retrieval performance. These findings indicate that teacher-child correspondence and behavioral phenotypes represent distinct forms of latent organization and demonstrate that the latent structure recovered from linked psychometric data depends on the representation learning objective.

Differential privacy provides a rigorous framework for protecting private information, typically achieved by adding random noise to query results. The generalized Gaussian family is a flexible class of additive noise distributions indexed by the shape parameter $p$ and includes the Laplace and Gaussian distributions as special cases $p=1$ and $p=2$, respectively. This paper studies the privacy-feasible scale estimation and the shape parameter selection of the generalized Gaussian mechanism (GGM) under $(\varepsilon,\delta)$-differential privacy. For a given sensitivity vector $\Delta$ and $p\in[1,\infty]$, let $b(p)$ denote the smallest value of the scale parameter for which the mechanism satisfies this privacy requirement. In the one-dimensional case, $b(p)$ can be implicitly characterized by a system of equations. For vector-valued queries, we construct a computable upper approximation of $b(p)$ that preserves the privacy guarantee. Shapes are compared under a scale-homogeneous utility criterion, with the $m$-th absolute moment as the main example. We develop an interval-wise shape search algorithm with an approximation guarantee that can be made arbitrarily precise. We also establish the invariance of the optimal shape under rescaling of the sensitivity vector and characterize its limiting behaviour under high privacy limits. Computational experiments show that optimizing shape parameters can improve utility by reducing the variance of each coordinate by 5% to 20% across a variety of cases, with some cases showing even greater reductions, while maintaining the same level of privacy protection. Task-specific experiments further show that shape optimization can improve task-level utility, reduce attacker success, or achieve both.

Calibration under model misspecification is inherently ill-posed because calibration parameters and structural discrepancy are statistically confounded without additional assumptions. Bayesian formulations address this ambiguity through prior and covariance modeling choices, including multi-response observations and cross-source alignment. However, how these choices, together with output-dependent uncertainty, shape posterior geometry and practical parameter identifiability remains poorly understood. We present a latent-variable multi-response calibration model that combines empirical Bayes estimation of nuisance hyperparameters with a Fisher-information-based local Gaussian approximation of the calibration-parameter posterior. Closed-form expressions are derived for squared-exponential and Matérn covariance functions, revealing how multi-response correlations, latent alignment, and output-specific uncertainty influence local posterior geometry. Controlled numerical experiments show that additional responses provide complementary geometric constraints on calibration parameters, while noise modeling can improve uncertainty quantification by reducing overconfidence without necessarily improving parameter localization. A lithium-ion battery study further illustrates how the model distinguishes strongly from weakly constrained parameters. Together, these results establish expected-Fisher posterior geometry as a scalable diagnostic of local parameter identifiability in multi-response calibration.

Natural-language tasks can elicit different verdicts from protocol-following evaluators that receive the same declared information. We study aggregate disambiguation systems (ADSs). Given a task and a candidate solution, each evaluator casts a binary vote on whether the solution should be accepted, and the system aggregates the votes of a finite panel. The target is protocol reproducibility relative to an explicitly declared evaluator reference, not semantic truth. We separate fixed finite censuses, probabilistic evaluator populations, and growing-census limits, since their endpoint laws and guarantees are not interchangeable. In the population setting, we use finite samples to estimate how often a finite panel reaches the same decision as the declared evaluator population. We provide a lower confidence bound on the fraction of candidate solutions for which the disagreement probability is at most a chosen tolerance. The calculation accounts separately for sampling candidate solutions and sampling evaluators. The construction permits arbitrary dependence among columns induced by shared evaluator rows and uses exact binomial intervals at the evaluator layer and an exact one-sided binomial inversion at the generator layer. Simulations check the implementation against known population coverages and expose power limitations.

Clinical validation studies of predictive tests are usually designed to focus on sensitivity ($Se$) and specificity ($Sp$), while statistical power is often calculated on regression-effect scales (e.g., risk ratio, hazard ratio). However, these quantities are statistically connected. Here, we provide closed-form links from sensitivity, specificity, and disease prevalence ($\pi$) to predictive risks, risk contrasts, and Wald-type variance, power, and sample-size formulas for binary and fixed-horizon survival outcomes. Analyses of statistical efficiency via C- and D-optimal principles demonstrate how prevalence and threshold choices affect study efficiency, supporting rapid decisions in preliminary studies and informing the design of subsequent, larger studies. Simulations show good calibration across most realistic scenarios; when events are rare and test effects are simultaneously very large, continuity and minimum-event corrections are needed to stabilize the approximation. We illustrate the framework with a case study describing use of the coronary artery calcium score for predicting incident cardiovascular disease in patients with type 2 diabetes mellitus. The formulas let investigators check power and required enrollment directly from $(Se,Sp,\pi)$, without running a separate simulation for each design candidate.

We develop a marginal coordinate test for regression with Euclidean predictors and a random-object response in a separable metric space. The goal is to test whether a predictor provides additional information about the response conditional on the remaining predictors. In a semi-supervised design, an unlabeled sample is used to estimate predictor conditional means, while an independent labeled sample is reserved for inference. The resulting residuals are combined with a product-space kernel to form a kernel conditional mean dependence (KCMD) U-statistic without requiring a response residual. The primary identity-based test targets a necessary conditional mean restriction, while a multiple-transformation extension probes broader alternatives. We establish a weighted centered chi-square null limit, wild bootstrap validity, consistency against fixed detectable alternatives, and local power under mean-element alternatives. For simultaneous inference, truncated p-to-e calibration combined with e-BH provides asymptotic false discovery rate control under general dependence. Simulations with Euclidean and non-Euclidean responses, together with a New York City taxi-flow analysis, illustrate the method.

Machine learning systems are increasingly corrected while they run, and the decision of when to intervene is increasingly delegated to statistical monitors. Anytime-valid inference promises evidence that can be acted on at any moment, exactly the guarantee this setting needs, and it is moving from theory into deployed monitoring. Conformal test martingales are the change-detection instrument, and Ville's inequality caps their false-alarm probability on exchangeable data. The guarantee is conditional. A deployment inherits it only if the stream it monitors behaves exchangeably. The premise is hardest to satisfy where these monitors are most useful, on dependent data and inside loops where the monitor modifies the learner whose scores it reads. It is also rarely measured. We measure it in a pre-specified case study, where such a monitor gates the online updates of a Kalman adapter correcting frozen time-series foundation models on five forecasting streams. On exchangeable synthetic streams, the same implementation fires in at most 1 of 60 runs. On the real streams, at alpha = 0.05, 135 of 135 clean-stream runs fired. The construction does not explain the firing; the failure comes from the deployed score stream itself. Repeated fires hold the gate's drift response active, and the gated filter amplifies the very transient it was designed to prevent. The component worth keeping makes no validity claim. Huber-style gating of the filter's own updates cuts isolated-spike degradation by an order of magnitude with no dataset specific tuning. Anytime-valid methods proposed for dependent data should therefore be accompanied by null-calibration controls and mechanism traces.

Mediation analyses are increasingly conducted in randomized trials, but a sample size adequate for the total treatment effect may leave the natural indirect effect (NIE) or natural direct effect (NDE) substantially underpowered. Randomization does not extend to the mediator, so precision depends on the conditional mediator distribution and the mediator-outcome association, neither of which enters a total-effect calculation. Planning outside linear structural equation models is largely based on simulation under a fully specified data-generating mechanism rarely available at the design stage. This paper develops analytic power and sample size formulas for the NIE and NDE with a binary mediator and a continuous or binary outcome. Under standard identification assumptions, we focus on the ratio-of-mediator-probability weighting (RMPW) estimator that does not require an outcome model for effect estimation. We decompose the oracle variances of the RMPW estimators into components capturing mediator-probability-ratio variability, outcome variation, and their association, with an additional shared-arm covariance term for the NIE. Under a probit latent-index mediator and a working outcome model, these components are determined by a small number of interpretable design inputs rather than by the full joint distribution of covariates, mediator, and outcome. Simulations show that the analytic sample sizes closely match simulation-based benchmarks, attain the target power, and maintain type I error near the nominal level. An ACTG175 illustration shows how pilot data can calibrate the inputs.

Studies with staggered entry, in which individuals enroll at different calendar times, are ubiquitous in medicine and related disciplines. Because these studies usually have a fixed administrative end of follow-up, identification of the estimand of interest relies on assumptions about the right-censoring mechanism. The assumptions are often considered plausible only conditional on covariates, including the time an individual entered the study (E). Yet, censoring assumptions formulated conditional on E are ill-posed due to positivity violations. Here, we study the consequences of this issue for common procedures in causal survival analysis, such as those based on marginal structural models. We further give conditions for valid identification and introduce sensitivity analyses to assess practical implications. We illustrate our methodology through two case studies. The first builds on the seminal article by Hernán et al. (2000) on the effect of zidovudine treatment. The second reanalyzes a recent study on the effect of the mRNA vaccine in patients receiving immune checkpoint inhibitors.

Researchers often summarize mediation in clustered data with a single-level product-of-coefficients estimator. We show what this pooled analysis estimates and why the target depends on the design. The pooled a-path is weighted by the within-between variance composition of the treatment. The pooled b-path uses a different weight, based on the mediator variance that remains after residualizing on treatment and covariates. Because these weights generally differ, the pooled indirect effect can differ from a level-respecting combination of the within- and between-level indirect effects. We derive the pooled probability limit and split this difference into a nonlinearity term and a weight-mismatch term. Equal b-paths remove the distortion, but equal a-paths do not. Allowing the two weights to vary gives a bilinear surface with saddle geometry, sharp corner bounds, and a zero-distortion frontier. We also define a design-standardized indirect effect and practical diagnostics. Simulations verify the identities and show that a common decompose-first, cluster-weighted construction can overstate apparent distortion. In an ECLS-K:2011 illustration with 8,363 children in 753 baseline schools, the within-level weights were 0.723 and 0.896, a difference of -0.173 (95% bootstrap CI [-0.193, -0.151]). The conventional construction produced 1.76 times the absolute distortion of the identity-consistent construction. The identity-consistent AD interval included zero. Thus, the application shows unequal weighting and sensitivity to construction, not evidence of nonzero population distortion.

Heavy-tailed $p$-value combination tests are attractive under unknown dependence because their null tails can be first-order robust even when the exact dependent null distribution is unavailable. That robustness does not resolve calibration: $\Pr\{T>q(\alpha)\}=\alpha+o(\alpha)$ neither quantifies the remaining size error nor determines its sign. We develop a second-order calibration theory for positive Half-Cauchy and reciprocal, or harmonic-mean, aggregation. After exact marginal standardization, extremal dependence is represented by an index-one exponent measure on the coordinate-face lattice. Its support separates axial, full-interior, proper-face, and mixed geometries. A Möbius decomposition shows how the noncompact weighted half-space probes every face, while quantitative face limits determine whether the correction is integrable, critically amplified, or nonintegrable. With common-factor inversion and a tail-to-calibration map, this yields machinery applicable beyond individual parametric copula families. The results include an integrable hidden-face transfer theorem, a sharp fixed-dimensional local-corner theorem with explicit higher-face control, a common-heavy-factor theorem, and size and critical-value expansions relative to independence. Gaussian, standard multivariate-$t$, positive Clayton, and max-linear pair-shock models exhibit distinct power, logarithmic, radial–angular, and proper-face mechanisms; the Gaussian weighted-half-space transfer is conditional on explicit face-boundary hypotheses. Thus dependence geometry determines the rate, coefficient, and direction of the calibration error left unresolved by first-order validity. Asymptotics use $t\to\infty$, equivalently vanishing significance levels.

Forecasting a correlation or covariance matrix is common in risk management and portfolio construction, but evaluating such a forecast correctly is not routine: naive matrix-comparison metrics are not proper scoring rules, walk-forward evaluation windows are easy to overlap with the estimation window in ways that silently leak information, and significance testing on serially dependent forecast-error sequences needs machinery few analysts implement from scratch. corrscore is a Python package that provides matrix-aware implementations of two established proper scoring rules for this setting – the energy score and the variogram score – dispatched across a closed-form tractability spectrum (point, discrete-mixture, and isotropic-Gaussian-mixture forecasts are scored exactly; a general Monte Carlo ensemble falls back to sampling), a geometry-aware variant of the variogram score built from the affine-invariant distance on the correlation manifold, a zero-overlap-by-construction walk-forward backtest harness, and a bundled significance-testing suite (circular block bootstrap, the Diebold-Mariano test, and the Model Confidence Set). We describe the package's design, its point of departure from the existing scoringRules and properscoring packages, and walk through a complete worked example.