Papers

Evaluating an AI system requires disaggregated assessment, as performance varies across domains such as benchmark task types or conversation types in deployed agents. Exhaustive testing is expensive, so evaluation rests on a sample of labeled units. We treat the evaluation set as a finite population and seek accurate point and interval estimates of each domain mean. Direct estimators, including prediction-powered inference (PPI), use only a domain's own labels and are imprecise where labels are few. Small area estimation addresses this problem, and we build on it to develop an integrated workflow for estimation and validation. For estimation, we propose prediction-powered smoothing (PP-S), a Bayesian model fit to each domain's prediction-powered estimate, with an extension that borrows strength across a reporting taxonomy (PP-TS). For validation, we derive a new, approximately unbiased design-based cross-validation score for choosing among direct and smoothed estimators. We study a curated benchmark with verifiable grading and deployed agent traffic graded by humans, each with every outcome observed. In both, the proposed estimators improve on the direct estimators in point and interval estimation, with near-nominal coverage. At the same sampling budget, our score selects as well as an independent validation sample does and estimates the selected estimator's error far more accurately.

Location estimation exhibits markedly different finite-sample behavior across noise distributions: regular families typically yield root-\(n\) rates, whereas compactly supported laws may admit faster, boundary-driven rates. We question whether a single estimator, without knowledge of the density's shape, can adapt to the instance-wise optimal estimation rate, as an oracle that knows the underlying location family can. For a known location family with symmetric log-concave noise density \(f\), the optimal location estimation error with sample size \(n\) under failure probability \(\delta\) is known to be Le Cam's two-point rate: \[ \sup\left\{r>0:\mathsf{H}^2\left(f_0, f_{2r}\right)\lesssim \frac{\log(1/\delta)}{n}\right\}. \] When the location family is unknown, we propose a shape-agnostic estimator that attains this oracle benchmark simultaneously over all symmetric unimodal densities with non-decreasing hazard rates, a class strictly broader than symmetric log-concave distributions. We establish that the Hellinger-driven two-point rate can be characterized solely by a multiscale function of dyadic quantile gaps. This new structural connection between Hellinger divergence and quantile geometry motivates a simple estimation procedure that aggregates sample mid-summaries with carefully designed data-dependent weights. The resulting estimator is finite-sample instance-optimal and runs in only \(O(\log(n))\) time on sorted samples.

Principal component analysis (PCA) is a fundamental tool for learning low-dimensional structure from high-dimensional data. When data are collected from multiple sources, the underlying task distributions may exhibit unknown degrees of similarity, with some tasks potentially arising from arbitrary distributions. We propose new multi-task PCA procedures that exploit similarity structure across tasks to improve eigenspace estimation while remaining robust to outlier tasks. We establish non-asymptotic convergence rates and show that the proposed procedures attain minimax optimal rates in a range of regimes. One of the procedures builds on the matrix-depth notion of Chen, Gao, and Ren (2018) and can achieve the optimal dependence of the estimation error on the proportion of outlier tasks, addressing a key challenge in robust multi-task learning. Extensive simulations and real-data analyses demonstrate the effectiveness of the proposed methods.

Likelihood evaluation in Bayesian inverse problems often requires a forward-model solve. We study candidate screening using a likelihood-weighted prior moment matrix and two associated scores: a Christoffel ratio and a Nevai polynomial average. A finite pilot supplies likelihood information that is reused to rank further prior candidates. We connect finite-pilot matrix error, quantitative Legendre localization, and fixed-budget posterior-mass regret. Conditional likelihoods distinguish feature loss from polynomial and sampling errors, and a filtered Gaussian construction permits controlled feature selection and localization. In a nonlinear function-coefficient PDE, equal-budget comparisons separate methods using scalar likelihoods from surrogates using forward outputs. The Christoffel ratio outperforms the tested direct likelihood regressions, but forward-response surrogates capture more posterior mass at small retention budgets. Independent-pilot tests and reference resampling quantify two distinct sources of uncertainty. Controlled Gaussian experiments illustrate finite-filter truncation and the benefit of selecting relevant features. These results identify when moment geometry provides useful screening information, without asserting universal superiority over surrogate models or an exact posterior sampling method.

Causal discovery from observational data is fundamental to statistics and machine learning, yet determining causal direction without interventions necessitates structural assumptions. Existing identifiability research primarily focuses on continuous variables under additive noise models, often neglecting mixed datasets containing ordinal scales, counts, and continuous measurements. This paper investigates causal discovery in Directed Acyclic Graphs (DAGs) where nodes follow either an ordinal distribution (via an ordered logit model) or a regular one-parameter exponential family distribution. We prove that the edge direction between an ordinal and an exponential family node is distributionally identifiable for generic parameter values. Our findings generalize previous Ordinal-Poisson results to the broader exponential family. Computationally, we introduce a score-based exhaustive search and a masked continuous optimization framework using DAGMA for larger graphs. Numerical results validate the theory, recovering edge orientations within a Markov equivalence class that are unidentifiable under classical structural equation models.

Many real-world experimental design problems require a batch of experimental runs across stages, in which multiple points are selected and evaluated at each stage. However, most work in the design literature is focused on fully sequential (point-by-point) methods. This paper proposes a sampling-based framework to systematically convert a fully sequential method to a batch sequential method. In particular, Stein variational gradient descent (SVGD) is adapted to efficiently sample a batch of points from a properly constructed target distribution while balancing the individual utility and the batch diversity. We address challenges that arise in using SVGD for experimental designs, including constrained design regions and near-uniform target distributions. We apply the proposed method to obtain batch versions of the state-of-the-art fully sequential methods, and demonstrate their performance through extensive numerical studies.

Classical confidence intervals, when repeatedly obtained on accumulating data at different sample sizes, produce contradictory inferences with high probability. We propose a simple and efficient strategy for computing, instead, mixture confidence sequences for regression coefficients in generalized linear models under this sequential framework. Simulations demonstrate the computational convenience of our approach and the importance of drawing inferential conclusions based on these anytime-valid tools when observations become available in batches over time. The usefulness of the proposed procedure is also shown in the analysis of streaming data from the American National Automotive Sampling System.

This paper develops inference for a Gaussian-nested hypergeometric family of distribution functions. The family \[ G_c(z) = \frac12 + z\,\frac{\Gamma(c-1/2)}{2\sqrt2\,\Gamma(c)}\,{}_1F_1\!\left(\frac12;c;-\frac{z^2}{2}\right),\quad c\ge\frac32, \] contains the standard normal distribution at the boundary $c=3/2$. Away from the boundary, the density has algebraic tail behaviour $g_c(z)\sim(c-3/2)|z|^{-3}$, so the parameter $c$ indexes a directed heavy-tailed deformation of the Gaussian law. The distribution also admits an equivalent beta-precision normal scale-mixture representation, in which the Gaussian boundary corresponds to degenerate unit precision. I establish the admissibility of the hypergeometric family, derive minimum-distance estimators, and obtain both regular interior asymptotics and nonstandard boundary asymptotics under the normal null. The standardised fit-improvement statistic converges to the mixture distribution $\tfrac12\delta_0+\tfrac12\chi_1^2$. I extend the theory to plug-in location-scale procedures, including a robust median/IQR version. Simulations document accurate null size and directed power against heavy-tailed alternatives. An application to daily S&P 500 returns, both unconditionally and after GARCH(1,1) filtering, illustrates the empirical implications for tail fitting and risk quantiles, and documents that GARCH filtering substantially reduces but does not eliminate the symmetric heavy-tailed departure detected by the test.

Time series of random objects, such as covariance matrices, probability distributions, and functional data, call for forecasting methods that do not rely on standard arithmetic operations. We introduce geodesic exponential smoothing, a generalization of exponential smoothing to time series in Hadamard spaces: the forecast level moves a fixed fraction of the way along the geodesic toward each new observation. The smoothing parameter is estimated by minimizing the average squared distance between observations and their forecasts. We further introduce an innovations mechanism under which each observation has conditional Fréchet mean equal to the current level, providing the metric-space analog of the innovations state-space model. In contrast to autoregressive models for object-valued time series, the framework involves a single scalar parameter, assumes no stationarity, and updates online in constant time per observation. Under this mechanism, we establish sample-path properties of the generative process via the quasilinearization available in Hadamard spaces, and prove almost-sure consistency of the smoothing-parameter estimator. Three real-data applications, spanning covariance-matrix, distributional, and functional time series, assess the forecasting performance of the method against structurally heavier alternatives.

Missing not at random (MNAR) data pose significant challenges for causal inference, particularly when both confounders and the outcome are partially observed. Without additional assumptions beyond those required for causal inference, causal estimands are generally not identifiable under MNAR mechanisms. This paper first develops identification results for the causal estimand, the average treatment effect, under several plausible MNAR mechanisms using completeness conditions, and proposes an estimation approach based on the Expectation-Maximization (EM) algorithm. We further extend this identification and estimation framework to mediation analysis, enabling the estimation of natural direct and indirect effects under MNAR mechanisms. Through extensive simulation studies, we compare the proposed method with two widely used approaches for handling missing data, complete-case analysis and multiple imputation. The results show that the proposed method yields substantially lower bias under the considered MNAR mechanisms. Finally, we apply the proposed approach to NHANES data to estimate the causal effect of education on depression, with health condition as the mediator.

Economic networks are often estimated separately over two periods, and changes in their edge sets are interpreted as structural rewiring. Since both networks are estimated, observed turnover also reflects graph-selection error. We study the two-snapshot Hamming-turnover functional under a homogeneous edge-misclassification model. With known sensitivity and specificity and conditional independence of the estimated edge indicators across periods, latent turnover admits a closed-form unbiased adjustment based only on observed turnover and the two estimated graph sizes. We then examine the effects of sparsity, calibration error and dependence across periods. When the number of true links is proportional to $p$, a false-positive probability of order $p^{-1}$ generates expected spurious turnover of order $p$. An error of the same order in calibrating the false-positive probability can likewise leave order-$p$ bias after adjustment. We also derive the bias induced by cross-period dependence and give sufficient conditions for consistency relative to network size. Numerical results illustrate the finite-sample implications.

This paper proposes an Expectation–Maximization algorithm with Metropolis–Hastings sampling for parameter estimation in a Mixed Fractional Merton Jump Diffusion model. The model combines fractional Brownian motion to capture long-range dependence with a compound Poisson jump process to describe abrupt movements in financial returns. The latent jump process is inferred during the E-step using Markov Chain Monte Carlo sampling, while the M-step updates the model parameters by maximizing the expected complete-data likelihood. The consistency and asymptotic normality of the proposed estimator are established under suitable regularity conditions. The methodology is applied to the Helsinki Stock Index (OMXH25). The proposed estimation framework provides a reliable and computationally efficient approach for modeling financial time series exhibiting both long-memory dependence and jump behavior.

Modern tabular datasets are becoming increasingly large, both in the number of samples and covariates, posing significant challenges for Bayesian variable selection due to the resulting computational burden. While there is extensive literature on scaling Bayesian inference to large numbers of observations or high-dimensional covariate spaces, comparatively little work addresses scalable Bayesian variable selection when both dimensions are large simultaneously in a practical setting. This paper presents a novel Bayesian variable selection framework for efficiently analysing data with a large number of both rows and columns. The proposed framework operates via a divide-and-conquer approach, splitting data into batches along both directions and analysing each batch independently in parallel, after which the results are combined together in a two-phase consensus procedure. Experiments demonstrate the computational gains while retaining strong variable selection performance, successfully identifying relevant signals in noisy, high-dimensional settings despite the loss of information induced by data splitting. Practical guidelines are also provided, including recommendations for tuning parameter choices and effective data partitioning strategies, followed by a real application to an H3N2 influenza dataset.

Graphical multiple testing procedures are a flexible and transparent way to control the family-wise error rate when a confirmatory trial pursues several label claims, but they leave open the question of which graph to use. In practice sponsors often fall back on fixed-sequence or Holm procedures that may poorly reflect what the trial is actually trying to achieve. We propose to instead choose the graph that maximises an explicit gain function, which states what each possible set of rejected hypotheses is worth to the sponsor: typically nothing until a regulatory hurdle is cleared, then an incremental amount for each further claim. Conventional power criteria are recovered as special cases, and because the search is confined to graphical procedures, family-wise error rate control holds whichever graph is selected. We set out a structured procedure for eliciting the gain function from the clinical, commercial, and regulatory members of a trial team. The proposed framework also handles uncertainty in the treatment effects and correlations assumed at the design stage, by averaging performance over their plausible values rather than fixing a single assumption. For group sequential designs, we describe how it can further reward early claims. Five examples, drawn from real and hypothetical pharmaceutical trials, show that the best graph depends on how trial success is defined, how uncertain the design assumptions are, and, in a group sequential setting, when claims can be established.

Phase-type (PH) distributions are versatile semi-parametric models for lifetime duration and can be used in survival and reliability analysis. In this paper we put forward methods and software for using PH distributions in a competing-risks model. The resulting multi-absorbing phase-type (MAPH) distribution records both the time until absorption and its cause. After formulating basic properties of this family, we develop an EM algorithm for parameter inference from exact event-time/cause observations and independently right-censored observations. For a censored subject, the E-step conditions on survival up to the censoring time and imputes both the latent transient path and its eventual cause of absorption; closed-form cause-resolved sufficient-statistic expectations are obtained from matrix exponentials. We illustrate applications and numerical properties and describe two accompanying Julia packages, in which both the exact-event and the censored E-step are implemented. Applied to intensive-care length-of-stay data in full, censored records included, the method attains a higher likelihood than the phase-type competing-risks fit previously published for those data.

Existing methods for identifying rapid-responding behaviour in large-scale assessments require parametric assumptions about the population. In this study, we propose a novel, non-parametric, mutual information-based framework of methods as an alternative. The methods within this framework compute the mutual information of the observed responses and discretised response times and maximise the information gain to determine a threshold that differentiates rapid responses from engaged responses. The only difference between the methods is the number of categories in the relevant variables. We present three methods explicitly. The first method uses response correctness and binarised response times. The second method uses correctness and categorises time into three groups. The third method uses raw responses and binarised times. We applied these methods to mathematics achievement data collected through the Programme for International Student Assessment in 2022. Furthermore, we examined the behaviour and usability of the first method in certain realistic conditions at the population and realised levels. The results indicated that the proposed framework is a viable alternative for identifying rapid responses. The framework's novelty lies in its non-parametric nature and its ability to utilise raw responses instead of correctness. Finally, we discuss some possible future research directions on this topic.

Matrix graphical models aim to characterize conditional dependence structures in matrix-variate data under a separable covariance assumption. In this framework, the precision matrix is decomposed as a Kronecker product, enabling separate modeling of undirected graphs across row and column domains. Existing methods have been developed for this problem, including likelihood-based approaches and regression-based procedures for graph estimation. Likelihood-based methods estimate precision matrices directly and recover graph structures indirectly, whereas regression-based approaches directly target estimating edges among variables, thus outperforming the former. However, existing regression-based methods are based on multiple penalized regression problems, which naturally yields asymmetry in estimated graphs and computational difficulty in selecting tuning parameters. To address the limitations, we propose a joint estimation of partial correlations in matrix graphical models. The proposed method estimates all partial correlations simultaneously within a unified optimization framework, thereby preserving symmetry and easing the pain of selecting the best models. Numerical studies demonstrate that the proposed method improves graph recovery performance compared to existing approaches. We also analyze protein expression data collected from patients with pulmonary tuberculosis, measured repeatedly at multiple time points, where the proposed method compares protein networks between two groups of patients and recovers the temporal dependence structure.

Several institutions test one family of hypotheses under family-wise error control but cannot pool their data, so each site releases only a quantized report of its p-value for each hypothesis, such as whether the p-value is significant. The report's resolution is the number of values it can take. We quantify the power lost to such reports relative to the most powerful centralized procedure given all p-values (the oracle), with reports and combining rule chosen optimally. A report from which the p-value can be recovered loses no power, so the loss is due to compression, not decentralization. Under the stated regularity and power-objective conditions, every finite resolution loses power, and the optimal loss decays as the inverse square of the resolution: equal-width intervals attain this order, and no partition into as many cells improves it. In two-site Beta examples, optimized one-bit reports are nearly lossless for a single hypothesis but lose several times as much power for two. At level 0.05 with two Beta(1,2) hypotheses, optimally combined three-bit equal-width reports retain 99.2% of oracle power. We give the power-optimal combining rule for given reports, and a rule using only their exactly known null distribution, valid under arbitrary dependence across hypotheses.

We study sparse composite quantile regression (CQR) for distributed data with heterogeneous honest sites and Byzantine workers. Honest sites share a common slope but may differ in their covariate distributions, error laws, and quantile intercepts. The proposed heterogeneity-calibrated robust CQR (HC-RCQR) profiles local intercepts and calibrates scores using an approximate inverse profile Hessian. Honest workers transmit the resulting vectors, whereas Byzantine workers may send arbitrary vectors. The server updates the estimate by coordinatewise trimming and soft thresholding. A scalar example shows how unequal honest-site curvatures allow intermediate Byzantine reports to survive trimming and how ideal calibration reduces their possible effect. We also establish nonidentification of the mean of unrestricted honest-site slopes when fault identities are unknown. Under suitable conditions, we establish conditional contraction and support-recovery guarantees. The bound separates score offset, sampling fluctuation, contamination, calibration error, and the Newton remainder. Simulations and a bike-demand study examine performance under heterogeneous data and adversarial messages.

Growth curve models are widely used in psychological research, and variable selection can help identify baseline characteristics associated with longitudinal heterogeneity. However, conventional inference after data-driven variable selection can be invalid because the same outcome data are used for both selection and inference. We develop a data-fission framework for post-selection inference in growth-curve models that separates the information used for selection and inference while retaining all participants in both stages through the addition and subtraction of Gaussian noise. The framework accommodates flexible variable-selection procedures and targets covariance-weighted linear projection parameters in the selected working model. It provides exact inference when the covariance structure is known, and we establish asymptotic validity under suitable regularity conditions when the covariance structure is estimated. Simulations show that the proposed method provides valid inference and improves efficiency over subject-level data splitting, whereas naïve post-selection inference can be biased. An application to the Longitudinal Study of American Youth illustrates the proposed framework.

This paper establishes a unified non-linear regularization framework for high-dimensional $M$-estimation, encompassing both linear models and Cox's proportional hazards models. Rather than relying on traditional additive non-convex penalties, the proposed paradigm embeds sparsity directly into the transformation for the physical parameter $\theta=\phi(\beta)$ using a smooth ($C^2$) component-wise "ReParametrization map for Sparsity (RePS)" $\phi$, and the penalty term is $\lambda \Vert \beta \Vert_1$ rather than $\lambda \Vert \theta \Vert_1$. This structural formulation dynamically adapts to local parameter scales, suppressing high-dimensional noise while simultaneously recovering unbiased oracle asymptotic normality under the large-sample limit. Through the Primal-Dual Witness method, we establish a unified oracle equivalence for both model classes under a liberated micro-penalty scaling regime $\lambda \ll n^{-1/2}$, accommodating model-specific structures such as the shift-invariance in survival analysis. Extensive Monte Carlo simulations demonstrate that the proposed framework consistently achieves superior false-positive control and high 95% confidence interval coverage in high-dimensional linear models, while also delivering performance comparable or superior in all aspects to state-of-the-art methods like SCAD and MCP in proportional hazards models.

Multivariate longitudinal data may exhibit non-Gaussian margins, nonlinear dynamics, and response vectors with composition that varies across waves. To account for these features, we introduce a vector drawable vine (VD-vine) copula that extends conventional drawable vine copulas from scalar to vector-valued nodes. Here, the response vector at each wave forms a multivariate marginal, and serial dependence is captured through a sequence of linking vector copulas. We establish that the VD-vine is itself a vector copula and reduces to a conventional drawable vine for scalar nodes. Recursive forward and backward conditional transports are derived that enable efficient likelihood evaluation and predictive simulation, with parsimonious reductions under finite-order Markov and stationary restrictions. Unconstrained parameterizations for Gaussian and FGM linking vector copulas, flexible multivariate marginals, and Bayesian variational inference provide a practical implementation. Simulations show improved predictive accuracy when the marginals are asymmetric and serial dependence is multivariate, with little loss under a correctly specified Gaussian panel vector autoregression. In an eight-wave Australian panel of 1,093 individuals with varying response vectors, the full VD-vine delivers the best cross-validated distributional forecasts among the models considered, establishing the benefit of capturing asymmetry and nonlinear dependence.

Active subspace methods provide a framework for quantifying predictive uncertainty in high-dimensional models by identifying and performing inference along parameter directions that have the greatest influence on the model output. However, the construction of active subspaces requires storing many full-dimensional model gradients, which becomes prohibitive as model size increases. We address this limitation by proposing Compressed Active Subspaces (CAS), a scalable approach that first maps the model parameters to a compressed space using a structured isometric embedding and then constructs the active subspace within this reduced parameterization. Our approach substantially reduces the memory required for active subspace construction and enables Bayesian inference for large models where standard active subspace methods become impractical. We demonstrate the scalability of CAS on neural networks of increasing size while maintaining predictive performance and robust uncertainty estimates.

Receiver Operating Characteristic (ROC) and Precision-Recall (PR) curves are widely used to assess the discrimination ability of forecasts for binary events, such as threshold exceedances or warnings of extreme events. In weather forecasting, forecasts are provided as spatial fields, yielding location-wise ROC and PR curves that are often aggregated to facilitate comparison. However, the effect of the aggregation strategy on performance assessment remains poorly understood. We investigate how different aggregation strategies for ROC and PR curves affect the assessment of discrimination ability. In particular, we identify conditions under which aggregation strategies satisfy two desirable properties for fair comparison: preservation of dominance between forecasts and preservation of concavity or achievability of the curves. We obtain sufficient conditions and propose two strategies satisfying them. They are compared with existing strategies from the literature, and we analyze their properties and highlight potential pitfalls that may lead to misleading interpretations. Based on these findings, we provide practical guidelines for the interpretation of aggregated ROC and PR curves. The proposed framework is illustrated with AI-based global weather forecasts, showing how different aggregation strategies can yield different rankings of competing forecasts.

Actor-partner interdependence models (APIMs) describe how one member's exposure is associated with that member's and the partner's outcomes, but actor and partner coefficients are causal only under additional design and identification assumptions. We formulate APIM-style questions as dyadic target trials under within-dyad interference and make three links between the associational and causal formulations explicit. First, under joint exchangeability, positivity, and a homogeneous correctly specified treatment-related conditional mean, APIM actor, partner, interaction, and joint coefficients equal the corresponding causal contrasts. Second, for binary dyadic treatment, multiplying the true member-specific propensities induces an exact one-parameter discrepancy delta(H) = Cov(A_1, A_2 | H): the four joint-cell errors are (+delta, -delta, -delta, +delta), total variation is 2|delta|, and IPW contrast bias has a closed-form representation. Third, the same discrepancy affects AIPW through its product with outcome-regression error. A 36-scenario simulation and a true-marginal factorization diagnostic show that large separate-propensity bias under residual dependence is primarily factorization bias, while flexible AIPW reduces nonlinear-model bias but not weak-support instability. In the strongest dependence condition, factorizing even the true marginals reversed the spillover contrast and inflated the interaction to nearly five times its true value. We apply the framework to 2,400 Health and Retirement Study couples studying retirement transitions and later depressive symptoms. Joint retirement was rare, residual treatment dependence was substantial, and estimates were sensitive to support and propensity flooring. The contribution is a design-and-diagnostic bridge from APIM questions to causal dyadic analyses, not a new general interference or AIPW theory.

In infectious disease transmission, regression and variable selection for time-to-event outcomes are complicated by transmission-induced dependence that violates standard independence assumptions. Pairwise survival analysis addresses this dependence by modeling contact intervals in ordered pairs, defined as the time from the onset of infectiousness in the pair's first individual to their first infectious contact with the second individual. We propose a penalized pairwise accelerated failure time model for contact intervals that accommodates simultaneous covariate effects on infectiousness and susceptibility while jointly representing within-population transmission and infection from external sources. For variable selection with many candidate covariates, we develop adaptive LASSO penalized maximum likelihood estimation with a coordinate descent algorithm, treating distributional baseline parameters as unpenalized nuisance parameters. We establish selection consistency and asymptotic normality of the adaptive LASSO estimator under profile-likelihood regularity conditions in a local asymptotic regime in which information accumulates over all ordered pairs that are ever at risk of transmission during follow-up. Simulation studies show accurate variable selection and improved estimation accuracy compared with the unpenalized estimator and the LASSO. We illustrate the method using 2009 household influenza A (H1N1) surveillance data from Los Angeles County, where the adaptive LASSO produces a sparse, interpretable model and improves the precision of selected covariate effect estimates, including the effect of antiviral prophylaxis on susceptibility.

To design many Bayesian studies, the sample size is chosen to attain sufficient power to reject a null hypothesis or a desired probability that an interval estimate is sufficiently narrow. Determining the minimum suitable sample size for studies with complex models typically requires intensive Monte Carlo simulation. One simulation-based strategy to consider the sample-size space involves estimating sampling distributions of posterior summaries, such as posterior probabilities or interval estimate lengths, at various sample sizes. Another strategy instead explores these sampling distributions via stochastic root finding over the sample-size space, where the entire sampling distribution of posterior summaries is not estimated at any given sample size. The performance of these competing approaches can be assessed using their Monte Carlo sampling distributions of sample size recommendations across repeated implementations. In this paper, we theoretically derive these Monte Carlo sampling distributions for various Bayesian sample size determination approaches. To promote best practices for Bayesian sample size determination, we conduct extensive numerical studies to estimate these Monte Carlo sampling distributions for the sampling distribution estimation and exploration approaches to assess their bias, precision, and sensitivity to tuning parameters.

We study the omitted variable bias (OVB) problem in canonical difference-in-differences (DiD) designs when unobserved confounding induces departures from the parallel trends assumption. Our results provide a novel characterization of the OVB formula for the average treatment effect on the treated (ATT), which is of independent interest. We show how the ATT bias is mainly governed by the strength of confounding in the treatment assignment mechanism and provide alternative ways of quantifying this strength, such as (i) changes in the average odds of treatment among the treated, (ii) confounding imbalance between treated and control units, or (iii) variation explained in treatment odds among the untreated. Building on these results, we offer sensitivity statistics for routine reporting, describing the minimum strength of confounding required to overturn the conclusions of a DiD study, as well as formal bounds on the strength of confounders based on comparisons to observed covariates or pre-trends. Finally, we provide flexible and efficient statistical inference methods for the bounds on ATT, which can leverage modern machine learning algorithms for estimation. We demonstrate the utility of our approach in an empirical example that estimates the effects of minimum wage on teen employment.

PURPOSE: Dynamic Bayesian borrowing methods adapt the contribution of external information according to agreement or disagreement with current data. Qualify-Then-Borrow (QTB) makes explicit how scientific source assessment determines what external information reaches the borrowing model and when observed outcome agreement should determine borrowing strength. METHODS: QTB uses five steps: define the target; assess the external data; classify them as qualified, repairable, or not qualified; use them directly, repair them before use, or exclude them; and then apply the selected Bayesian borrowing method. Dynamic borrowing is the final step and operates on residual empirical compatibility after identified material differences have been addressed. We evaluated 82 prespecified scenarios with 10,000 replications each. RESULTS: Outcome data alone could not determine the appropriate borrowing decision. Two settings with the same external outcome distribution produced similar robust-mixture behavior but different QTB decisions because one had a known endpoint problem. When misclassified data were borrowed, full pooling reduced 95% coverage to 60.8%; robust-mixture downweighting improved coverage to 91.8% but left a -4.8 percentage-point bias. Repair of a measured population shift reduced absolute bias from 1.94 to 0.80 percentage points. Under strong negative residual drift, Type I error reached 10.0% despite downweighting. With fully compatible data, QTB reduced to the usual robust-mixture analysis. CONCLUSION: QTB is a five-step decision architecture, not a new prior or compatibility statistic. It determines whether external information proceeds unchanged, requires repair, or is excluded before dynamic borrowing operates on residual empirical compatibility. Qualification does not certify exchangeability, and downweighting is a safeguard rather than a guarantee against residual bias.

In clinical settings, predicting binary outcomes is often complicated by right-censoring, which prevents us from distinguishing individuals in whom the event never occurs (i.e., negative, or non-susceptible) from those where it occurs after the censoring time (i.e., positive, or susceptible). To discriminate between these subpopulations, we can use mixture cure models (MCMs), which model both the outcome probability over an infinite horizon and its time-to-event (TTE) distribution when observed. However, the TTE approach requires event timestamps, which can be unreliable, costly to obtain, or biased due to disparities in clinical care. Alternatively, the task can be formulated as positive-unlabeled (PU) learning, in which individuals without an observed event are grouped as "unlabeled" rather than negative, recognizing that this group includes positives whose event occurred after censoring. Here we analyze the relationship between these frameworks and provide evidence-based recommendations for choosing between them. We begin by showing that both families optimize a shared likelihood but differ in how they constrain the probability of event occurrence prior to censoring among susceptible patients, which in PU learning is known as the labeling propensity; and that an MCM may be viewed as a PU model whose labeling propensity is parameterized as a TTE distribution, which makes the model identifiable. In systematic simulations and two real-world clinical cohorts, we examine whether and when including the TTE component benefits binary classification. We show that MCMs are preferred in only one specific regime: under selective censoring where the exact features governing censoring cannot be identified. Otherwise, PU learning achieves equivalent performance while avoiding challenges and potential biases associated with the TTE approach.