Papers

Let $X=(X_1,\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\ldots,D_n)\sim\operatorname{Dir}(1,\ldots,1)$ be independent of $X$, and define $K(x)=\mathbb{P}\{\sum_{i=1}^n x_iD_i\le1\}$. We prove that, for every $n\ge1$, whenever $\mathbb{E} X_i\le1$ for every $i$, $\mathbb{P}\{K(X)\le\alpha\}\le\alpha$ for all $0\le\alpha\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\mathbb{E}X_i \le 1$ for all $i$. This proves a conjecture of Gaffke (2005).

Backpropagation (BP) dominates deep learning training, but its reliance on gradients brings inherent troubles – vanishing and exploding gradients. The pursuit of gradient-free methods has long been a goal in the field of artificial intelligence. This paper shows that indeed the simplest Monte Carlo algorithm implemented on a single GPU – randomly mutate a parameter, keep it if the loss decreases, otherwise retry – can practically train deep networks. This gradient-free method does not even need common techniques such as batch normalization or residual connections to directly train sufficiently deep networks. More remarkably, its flexibility extends to several nontrivial scenarios: it enables pure pruning training, supports discrete weights, accommodates unconventional transfer functions such as Gaussian, and reveals the substantial redundancy of deep networks. We have demonstrated its feasibility on deep networks with more than 20 layers, single-hidden-layer wide networks with up to 16,384 hidden neurons, and even a simple Transformer architecture trained on both image classification (MNIST) and character-level language modeling (Tiny Shakespeare). This simple gradient-free method may offer a complementary perspective for understanding the self-organization and learning mechanisms of neural networks, and also provides an alternative route for building physically inspired deep learning systems.

Separability is an important structural assumption often placed on the covariance when working with matrix-variate data, because it greatly simplifies both interpretation and computation of subsequent covariance-based statistical tasks. Yet testing the separability assumption is difficult in the high-dimensional regime. We propose to test separability by recasting the problem as a sphericity test after whitening the data using the separable maximum likelihood estimate of the covariance. The test is calibrated by Monte Carlo simulation, yielding finite-sample level control. Furthermore, we prove the test's high-dimensional consistency under dense alternatives. To reduce its reliance on distributional assumptions, we introduce an angular version of the test based on radial normalization after whitening. We demonstrate the practical utility, empirical power, and computational efficiency of the prop

Distribution goodness-of-fit testing has developed a powerful rate-level theory: we often know how the required sample size scales with the alphabet size, the separation from the null, and the target error probability. Uniformity testing is the canonical example. One can distinguish the uniform distribution on $N$ categories from alternatives at total-variation distance at least $\epsilon$ with far fewer than $N$ samples, and the optimal scaling is now well understood. But rate-level theory leaves an important question unresolved: among several tests with the same sample-complexity order, which one actually gives the best risk or power? This is a constant-level question. It is especially relevant in modern applications where distribution testing is used not merely as an asymptotic abstraction, but as a practical design tool. This note argues that sharp constants in distribution testing play a role analogous to Fisher information in parametric estimation and Pinsker's constant in nonparametric estimation. First, they distinguish between tests that are all rate-optimal but not equally powerful. Second, they reveal the effective signal-to-noise ratio governing the testing problem. Third, they can guide tuning-parameter choices in downstream applications. We illustrate this perspective through large-alphabet uniformity testing and then explain why the same logic matters for choosing the number of bins in calibration testing.

Active testing provides a label–efficient approach to risk estimation by adaptively selecting which test points should be labelled. However, existing estimators fail to exploit the informative predictions of powerful black–box models, even though such predictions are increasingly available in settings where labels remain expensive. To address this, we propose Prediction–Powered Active Testing (PPAT), a novel label–efficient risk estimation framework that combines the unbiased LURE estimator \citep{farquhar2021statistical} with a prediction–powered control variate. Rather than using proxy predictions as biased pseudo–labels, PPAT uses them to residualise the loss, preserving unbiasedness while reducing variance. Beyond the estimator itself, PPAT also changes which points should be acquired: we derive oracle and practical surrogate–based acquisition rules tailored to reducing the variance of our estimator. Moreover, we establish asymptotic normality for PPAT, yielding asymptotically valid confidence intervals and thus a principled estimate of the uncertainty around our estimates. Across tabular regression and image–classification tasks, PPAT outperforms existing methods in risk estimation, while its confidence intervals attain the target coverage with substantially fewer labels and smaller widths.

Diffusion unlearning is essential for mitigating the generation of harmful or copyrighted content in text-to-image models. Current diffusion unlearning techniques determine the model update direction by either using alternatives of the target concept as an anchor or using empty prompts. The anchor-based method relies on manually and semantically-chosen anchors that risk biased unlearning, while the anchor-free method inherently suffers from unrobust unlearning due to unconstrained latent updates. In this work, we theoretically formalize such unstable diffusion unlearning issues under the manifold hypothesis and prove that lacking a manifold-proximal anchor inevitably induces significant normal-space drift that degrades unlearning performance. To achieve stable unlearning, we propose \mysysn, a two-stage framework that automatically synthesizes manifold-proximal anchors. However, direct geometric manifold optimization is computationally intractable. To address this challenge, \mysys introduces a novel cross-attention consistency loss which serves as a highly efficient surrogate of manifold proximity. Experimental results demonstrate that \mysys effectively achieves robust and unbiased unlearning across various state-of-the-art baselines, significantly improving targeted concept removal (by up to 31.04% in CLIP score) and non-target utility (by up to 4.18% in CLIP score). Moreover, \mysys can also be easily integrated into existing diffusion unlearning methods to enhance their unlearning performance (by 6.30% for concept removal and 6.65% for utility on average).

Policy-based approaches to Bayesian experimental design (BED) allow the learning of deep policy networks that adaptively make intelligent design decisions based on previously collected data. However, the training of such policies is often held back by a fundamental challenge: the double intractability of the expected information gain (EIG). This necessitates expensive or complex approximations that restrict the effort one can invest in optimising the policy itself. To address this, we show that the double intractability of the EIG can be isolated from the policy learning by first solving a score matching problem that is independent of the policy used, then using the learned score approximation to train the policy in a singly intractable manner. This turns the key multiplicative cost into an additive one and reduces the computational burden on the policy training itself, making it far cheaper to train the policy multiple times when needed, e.g. for architecture search, hyperparameter tuning, or avoiding local optima. In our experiments we train multiple competitive policies without inducing a multiplicative cost in likelihood evaluations, which can increase performance by allowing us to select the best policy even without performing hyperparameter or architecture searches.

Distance-based classifiers are very popular, and the Euclidean distance is one of the most commonly used metrics in distance-based classifiers. However, classifiers based on the Euclidean distance often suffer in high-dimensional setups due to issues such as distance concentration, violation of neighborhood structures, and the presence of hubs. In high-dimension, low-sample-size (HDLSS) situations, a data-driven semi-metric called the Mean Absolute Difference of Distances (MADD) is known to circumvent these issues. But one major problem with MADD is that its computational complexity increases quadratically with the training sample size. As a result, the application of MADD becomes computationally challenging for big datasets that have both a high dimension as well as a large number of observations. In this paper, we propose a scalable version of MADD that significantly reduces its computational complexity while retaining its advantages. This speed-up is achieved by selecting a representative set during the computation of MADD. Further speed-ups are achieved by using the idea of Random Fourier Features, particularly when the sample size is very large. We establish that our proposed methods achieve performances similar to MADD but only at a fraction of its computing time, both theoretically as well as numerically. Our approach broadens the scope of MADD, allowing its use to big-data with a very large number of observations.

In this work, we introduce circular expectiles as minimizers of an asymmetric circular loss function based on chord distance. In contrast to the linear expectile criterion, the resulting circular optimization problem is non-convex, so existence and uniqueness require a separate analysis. The construction extends linear expectiles to directional data while preserving the circular mean as the symmetric case corresponding to $\alpha=1/2$. We derive basic representations of the objective function and the associated identification function, and give a geometric interpretation that generalizes the corresponding representation for the circular mean. Furthermore, we prove the existence and uniqueness of the minimizers for distributions with positive density on the circle. The empirical circular expectile is defined by using the sample circular mean as reference direction for the induced linear order on the circle. We prove the uniqueness of the empirical expectile, as well as its consistency and finite-dimensional asymptotic normality. Finally, we indicate possible applications to circular measures of dispersion, skewness, and symmetry diagnostics.

Accurate poverty mapping using satellite imagery is often hindered by (i) noisy and sparse survey-derived supervision, (ii) image quality issues such as cloud cover and image corruption, and (iii) lack of explicit spatial structure in image-only models. Building on the KidSat framework, we develop an enhanced pipeline that improves predictive accuracy via refined data preprocessing, systematic image quality assessment, and mathematically defined geographic encoding. First, we refine the fine-tuning target matrix by resolving high-cardinality sparsity and reducing one-hot dimensionality from 103 to 51 via DHS re-aggregation. Second, we introduce a simple two-stage quality-screening procedure to filter heavily clouded or corrupted observations. Third, we fuse DINOv2 visual embeddings with Spherical Harmonics (SH) location features. Across extensive experiments, these changes reduce MAE from 0.2167 to 0.1759, corresponding to an 18.83% relative reduction on the cluster-level severe-deprivation proportion scale. When extended from 16 to 33 African countries, the best-performing configuration achieves an overall MAE of 0.1658. We find that SH features consistently improve performance over the image-only backbone, whereas higher-capacity coordinate Multi Layer Perception augmentation (SH+SIREN) can underperform without carefully designed objectives. Finally, gradient-boosted tree heads (XGBoost/LightGBM) most effectively exploit nonlinear interactions in the fused visual-geographic representation. These findings provide a scalable and principled recipe for improving satellite-based socioeconomic predictions using only publicly accessible data.

The R package glmSTARMA implements autoregressive models for spatio-temporal data at fixed locations, with time-invariant spatial dependency structure. We rely on generalized linear models methodology and unify several approaches for the analysis of spatial count time series. Such models allow the (conditional) mean of the response to depend on past observations, lagged (conditional) expectations, and covariates. The response can be a continuous or a discrete random variable. Additionally, the package develops inference for double generalized linear models, allowing the dispersion parameter(s) of the marginal distributions to be modeled similarly to the mean process. This is a new capability which introduces, for example, spatio-temporal volatility models, such as space-time GARCH processes, and count time series models with spatio-temporal overdispersion and underdispersion. We provide functions for model estimation, simulation, inference, and prediction. Its use is illustrated by data examples.

The k-means++ algorithm is commonly restarted multiple times to avoid poor local optima, yet the number of restarts is almost always chosen arbitrarily and applied uniformly regardless of data set difficulty. This undermines any comparison relying on such a choice and wastes computation on easy data sets while potentially under-serving hard ones. Here, we introduce the Good-Turing Restart Criterion (GTRC). This combines a Good-Turing estimate, a proven unconditional bound, and a confidence-based bound on the probability that a further restart would improve on the current result, stopping once this probability falls below a user-specified tolerance. Our experiments on 34 real-world data sets show that GTRC identifies the point beyond which further k-means++ restarts yield only negligible improvement, achieving a more favourable balance between the number of restarts used and clustering quality than three existing stopping rules for multistart local search and popular fixed restart counts. Software: https://github.com/RCdeAmorim/Good-Turing-Restart-Criterion.

Recent algorithmic advances have made directed acyclic graph (DAG) structure learning scalable for causal discovery. Yet, the currently available techniques assume a completely homogeneous population, precluding their application to clustered data where cluster-specific variations (e.g., patient-specific effects) are common. We address this issue by introducing a new approach that estimates a global structure while accounting for local cluster-level effects. The key idea is to extend the fixed- and random-effects framework of classical mixed models to the structure learning setting. Towards this end, we present a differentiable graph coupling mechanism that guarantees the union of the fixed- and random-effects graphs remains acyclic. Computationally, we provide a provably convergent first-order method and leverage efficient batched updates across clusters. Statistically, we establish identifiability of the model and show that our approach recovers the true structure asymptotically. In experiments on real and synthetic data, our proposal detects dependencies missed by alternative estimators, underscoring its value for structure learning in clustered settings.

Circular coordinates obtained from persistent cohomology reveal loop structure in data, but they usually remain abstract: A detected circle does not tell us which measured angle, phase, torsion, or decoder explains it. We propose a method for selecting interpretable circle-valued coordinates from a user-supplied dictionary of scientifically meaningful candidates explaining the detected cohomology. In the continuous setting, each candidate is represented by the cohomology class of its pulled-back angular form, and selecting a minimum-energy set of candidates spanning the relevant $H^1$ subspace becomes a minimum-weight basis problem in a vector matroid. We then introduce CIRCOL, a method for discrete point clouds sampled from the manifold. We prove that the introduced cochain inner product is a consistent estimator of the $L^2$ inner product of fixed smooth 1-forms under non-uniform sampling. The resulting projection matrix both helps selecting a basis of low-energy dictionary coordinates and diagnoses topologically trivial candidates or unexplained persistent classes. Finally, we verify the effectiveness of our method on synthetic examples, on molecular simulations, and neural recordings of head-direction cells.

In this paper, we study an age-dependent branching process. In the simplest setting, the population is divided into two age groups, namely juveniles and adults. Our objective is to estimate the model parameters using observations of the total population size only (i.e., juveniles plus adults). Focusing on the ergodic regime of the model, we introduce a method-of-moments estimator and establish its asymptotic normality. Several extensions are discussed, including models with more than two age groups.

The Synthetic Control Method (SCM) is a widely used approach for assessing the effects of interventions by constructing a synthetic counterfactual using a donor set of untreated units. However, the effectiveness of SCM heavily relies on the careful selection of an appropriate donor set. In this paper, we propose a Bayesian hierarchical model that performs donor set selection while preserving the standard SCM simplex constraint on donor weights. Unlike approaches that assume a fixed donor set, our model allows for the simultaneous estimation of the synthetic control weights and the active donor set. By using a hierarchical Gamma-Bernoulli construction for the donor weights, the proposed model assigns posterior mass to simplex faces and allows exact zero weights for excluded donors. We establish a posterior donor-set consistency result under a simplified pre-intervention model. Through numerical simulations, we show that our model improves donor recovery and weight estimation when the donor pool contains irrelevant or weakly related units, while remaining competitive in full-donor settings. Finally, we apply our model to the GDP trajectory of West Germany, illustrating its practical applicability. Our findings suggest that incorporating donor set selection offers a more parsimonious and flexible extension of existing Bayesian synthetic control methods.

Statistical analysis of high-dimensional data is often hampered by limited sample sizes, yet auxiliary datasets from related sources are often readily available. When two such datasets share part of their covariance structure, but not all of it, exploiting the shared part can substantially improve estimation. We propose a spiked covariance model that explicitly captures this partial sharing: two datasets share a subspace of unknown rank and arbitrary position in the spectrum, while each retains its own distinct spiked directions. The model treats the two datasets symmetrically and strictly generalizes existing models for shared covariance structure. We develop a complete estimation procedure that includes joint estimation of the shared subspace and its rank, a closed-form pooling weight for combining the two datasets, and asymptotic guarantees derived from random matrix theory in the proportional-growth regime. The framework also resolves a gap in contrastive dimension reduction by providing a principled estimator for high-dimensional settings. We illustrate the methodology on portfolio construction during the early COVID-19 pandemic and on contrastive analysis of brain tumor gene expression.

Inference for spectral edges of large covariance matrices is a fundamental problem in high-dimensional statistics. A major difficulty is that the largest non-spiked sample eigenvalues, which serve as natural estimators of the edge, fluctuate on the Tracy–Widom scale. Consequently, valid inference requires accurate centering by the deterministic spectral edge together with a precise scaling constant, both of which are often difficult to estimate in practice under general unknown population covariance structures. In this paper, we propose a bias-corrected multiplier bootstrap procedure for inference on the deterministic edge of the bulk spectrum. The key idea is to introduce a carefully calibrated multiplier perturbation that regularizes the edge fluctuation to a slightly larger scale at which Gaussian approximation becomes tractable. The resulting confidence interval is constructed directly from bootstrap eigenvalues, together with a data-driven recentering step that corrects the bootstrap-induced shift of the deterministic edge. On the theoretical side, we show that, after bias correction and rescaling, the largest few non-spiked bootstrap eigenvalues are asymptotically Gaussian conditionally on the data. Building on this result, we establish the asymptotic validity of the proposed confidence interval, whose length is only slightly larger than the Tracy–Widom scale, and prove vanishing coverage under alternatives in which additional spikes separate from the bulk at a local scale larger than $n^{-1/6}$. As a consequence, the same confidence interval yields a threshold-free estimator for the number of spikes, without requiring the spikes to be distinct or very large. Equivalently, the procedure yields a data-driven and theoretically justified cutoff for the scree plot.

Radiomic features derived from medical images and segmentation masks are used to support decision making in clinical imaging pipelines. In practice, these features are often computed from predicted masks, but segmentation models can be overconfident or poorly calibrated, making derived measurements appear more reliable than they are. Conformal prediction (CP) provides distribution-free prediction intervals with finite-sample marginal coverage guarantees, but black-box intervals for segmentation-derived radiomics can be inefficient because they ignore test-time information about image appearance, mask geometry, and segmentation uncertainty. We propose ConRad, a conformal framework for scalar radiomic targets that uses covariates derived from the predicted mask, input image, predicted radiomics, and boundary uncertainty to construct adaptive intervals while maintaining coverage. Across five 2D medical imaging datasets and 171 retained radiomic targets, we show that ConRad improves feature-level efficiency compared to baselines while maintaining near-nominal empirical coverage. Ablation results further indicate that segmentation boundary uncertainty features are the largest contributors to interval efficiency.

Combination therapy, a treatment approach that involves two or more monotherapies, is widely considered to enhance therapeutic efficacy across different medical conditions. It was often believed that combination therapy is beneficial because of pharmacological interactions between its component monotherapies. However, through laboratory experiments, pharmacologists have recently noted that the benefits of some combination therapies might be largely driven by varying patient-level responses to their component monotherapies. Without accounting for such patient-level heterogeneity, classical statistical inference frameworks for combination therapy might be inadequate and overly optimistic. In this paper, we introduce a novel and model-free statistical inference framework to complement the classical one and evaluate combination therapy after adjusting for patient-level heterogeneity in responses to monotherapies. We address the non-identifiability and nonlinearity issue inherent in adjustment of patient-level heterogeneity and establish conditions for the (partial) identifiability of the cross-world target parameter. We develop an outcome-based optimal matching scheme to achieve asymptotic normality and construct $\sqrt{N}$-rate confidence intervals for the target parameter, thereby enabling reliable, efficient and transparent evaluation of combination therapy amid patient-level heterogeneity. The benefits of the proposed framework are demonstrated through a reanalysis of the ACTG 175 trial.

Diffusion Language Models (DLMs) have recently achieved substantial progress in natural language generation tasks. Recent research demonstrates that adaptive token generation ordering can significantly improve performance in mathematical reasoning and code synthesis applications. In this work, we investigate the optimization of generation order for both text-to-image synthesis and multimodal understanding. We first establish that, unlike structured problems in language generation such as Sudoku puzzles, model logits alone are insufficient for determining optimal generation sequences in text-to-image generation and multimodal understanding. To address this challenge, we introduce a learnable control module trained via Group Relative Policy Optimization (GRPO) to determine the generation order. Our results demonstrate that learning this control block substantially improves both text-to-image alignment and multimodal understanding in DLMs. In particular, it enhances the model's ability to capture fine-grained spatial relationships in generated images while also strengthening performance on multimodal reasoning and comprehension tasks. We evaluate our framework on GenEval, an object-focused benchmark for text-to-image alignment, where it achieves 4.08% relative improvements. In addition, experiments on VLMEvalKit confirm 4.85% relative improvements in multimodal understanding, highlighting the broad effectiveness of our approach.

This paper develops principled updating rules for possibilistic inference, where uncertainty about a fixed parameter is represented by a possibility function, the maxitive analogue of a probability distribution, and comparisons are made pointwise via a partial order. From two complementary foundations, an information-conservation viewpoint and an axiomatic viewpoint, we derive the same canonical update: the posterior is the prior-likelihood product followed by supremum normalisation. The two derivations agree for an arbitrary loss, differing only in where the learning-rate parameter enters. This parameter controls epistemic strength and is not identifiable from the normalising evidence alone, clarifying the role of analogous learning-rate parameters in generalised Bayesian updating.

We combine neural network means with exact Matérn kernel regressions of their residuals and of their learned features, and evaluate the pairing on two public emulation problems with published baselines: the structural-mechanics benchmark of de Hoop et al. and the OCO-2 radiative-transfer emulator of Lamminpää et al. On structural mechanics the combination reaches 4.55% test error, matching the best published architecture, and 5.38% against a published 6.49% in the low-data regime. On OCO-2 it improves on the published Gaussian-process emulator on that problem's own test points, outright on two of the three spectral bands; the same kernel that trails the network tenfold on the raw state overtakes it on the network's features, and we measure why (the target's squared native-space norm drops about fortyfold at fixed effective dimension) and prove the mechanism. Where the two families tie instead, the residuals of every architecture we train correlate above 0.86 and their shared component is flat in diversity and sample size, which reads the published plateau as a property of the data. Supporting results include a second-moment identity that predicts stacking outcomes from measured correlations, an optimal-recovery certificate, and a distribution-free coverage band, the only uncertainty signal that survives our tests.

Evaluating decisions made under uncertainty is hard when labeled outcomes are scarce, costly, or confounded with luck. We treat subjective expected utility (SEU) maximization as a stated standard and define a graded measure – SEU sensitivity – of an agent's conformity to it. The vehicle is a softmax choice model with a sensitivity parameter $\alpha$ on SEU-valued alternatives; the contribution is a sequence of identifiability results for $\alpha$ and for belief and utility parameters $(\beta, \delta)$, validated in Stan via prior predictive checks, parameter recovery, and simulation-based calibration (SBC), with finite-sample caveats intact. In the uncertain-choice-only model $m_0$, $\alpha$ is identifiable given the expected-utility vector $\eta$ and sharply recovered, while $(\beta, \delta)$ are only weakly informed: the posterior barely contracts and concentrates on a $\beta$-$\delta$ trade-off. In the extended model $m_1$, $\delta$ becomes identifiable in principle via a $\beta$-free risky block, but its practical recovery gain at realistic sample sizes is negligible (matched-count CI-width reduction under 1%), and that block yields no detected $\alpha$-precision gain at matched choice count. These are two distinct phenomena: for $\delta$, identifiability does not imply precise estimability at realistic $n$; for $\alpha$, identifiability is silent about what governs finite-$n$ precision. Marginal SBC passes for both models even where the joint posterior is weakly informed – a demarcation we make precise. A two-by-two application (GPT-4o and Claude 3.5 Sonnet, each on insurance-claims triage and Ellsberg-style urns, with sampling temperature as the lever) runs end-to-end on real LLM choice data, detecting a structured comparative $\alpha$ effect in two of four cells.

Diffusion-based policies have recently emerged as powerful policy parameterizations for reinforcement learning, representing state-conditioned action distributions as terminal laws of diffusion processes with parameterized drifts. This terminal-law representation has shown substantial expressive flexibility in practice, enabling diffusion policies to model complex, multimodal, and highly non-Gaussian action distributions; however, it remains unclear what mathematically drives this expressivity and how to fully exploit it when the policy is learned from finite data. In this paper, we identify the drift Lipschitz budget $K$ as a central quantity governing the expressivity and statistical behavior of diffusion policies. We quantify expressivity through approximation: diffusion policies with $K$-Lipschitz drifts can concentrate near optimal deterministic policies and achieve value approximation error of order $1/K$; moreover, we prove a matching lower bound under nondegenerate diffusion noise. This increased expressivity comes with a statistical cost. When the drift is parameterized by neural networks, increasing $K$ improves approximation but increases statistical complexity. Balancing these two terms yields a finite-sample performance gap of order $\tilde{O}(n^{-2/(m+6)})$ for generic neural-network drifts, and a sharper rate $\tilde{O}(n^{-2/(m+4)})$ for one-sided dissipative drift classes, where $n$ is the sample size and $m$ is the dimension of the state space. Numerical experiments provide empirical evidence for the sample-dependent trade-off in $K$, supporting both theoretical regimes. Our framework also suggests a practical implementation principle: choose the diffusion budget $K$ according to the available sample size, and then select a neural-network architecture with the corresponding fixed Lipschitz coefficient.

How readily people override intuitive conclusions through reflection shapes how they navigate dense information environments with reliable and misleading sources; yet the effectiveness of a prominent measure, the Cognitive Reflection Test (CRT), is eroded by widespread exposure to classic items and leaves open how such tendencies manifest more generally in decision style and linguistic expression. The Behavioural Reflection Test (BRT) addresses these issues with a brief open-ended measure of reasoning in morally and epistemically charged scenarios, alongside a four-item bespoke CRT (bCRT) as a low-exposure anchor. Among 473 online adults, higher bCRT predicted more evidence-sensitive, ethically driven decisions and reliance on high-quality sources, marked by more emotionally engaged, risk-attentive, economical language; associations the familiarity-adjusted CRT did not recover. The bCRT showed convergent validity, added item information above mean ability. Though open-ended, the BRT remained a time-efficient (median 11.8 minutes) behavioural assay of reflection with scope to extend across domains.

Pairwise comparisons are fundamental in the analytic hierarchy process. Various consistency indices have been proposed to assess inconsistencies in these comparisons. Since Saaty first proposed his consistency index, the assessment of the degree of consistency in pairwise comparison matrices has remained an open and hot topic in the study of the analytic hierarchy process. The consistency indices CI and CR proposed by Saaty are defined using the principal eigenvalue of the pairwise comparison matrix. In our previous study, we introduced an alternative index derived from the relationship between the coefficient of the characteristic polynomial and the consistency of comparisons. Saaty proposed a fixed threshold of 0.1 for CI or CR as a guideline for an acceptable level of consistency, regardless of the matrix size. However, whether this threshold represents an equivalent level of consistency across different matrix sizes, that is, across different numbers of evaluation items, remains unclear. This study analysed the relationship between consistency and matrix size by examining pairwise comparison matrices constructed from subsets of evaluation items. Based on this analysis, we propose the fundamental property to be satisfied by a size-independent consistency index. Furthermore, we refine our previously proposed index to ensure that it satisfies this property, demonstrating that it coincides with the existing consistency index. Finally, we visualise the relationship between the matrix size and consistency index values using randomly generated pairwise comparison matrices, thereby providing insights into the impact of matrix size on consistency evaluation.

The COVID-19 pandemic produced a modest wave of college closures and mergers that may offer an early, if imperfect, preview of the demographic "enrollment cliff" anticipated in the coming decade. This paper examines the institutions that closed or merged between 2020 and the end of 2025. We assemble a dataset of 65 such institutions, pairing institutional characteristics with state- and regional-level demographic, economic, and financial indicators, and supplement it with a corpus of news coverage of the closures. Using a combination of Bayesian models, dimensionality reduction, clustering, and topic modeling, we describe where these closures occurred, what the closed institutions had in common, and how they were discussed publicly. Consistent with prior demographic projections, closures were more frequent in the Northeast and Midwest, though the absolute numbers remain small. The closed institutions were heterogeneous rather than uniform: financial structure, regional demographics, and institutional mission each contributed to distinguishing them, and religious affiliation recurred prominently in media coverage. We frame these results as exploratory and descriptive given the small sample, and we discuss what they may, and may not, imply for institutions navigating the enrollment cliff.

Interrupted time series analyses (ITSA) of proportion and rate outcomes are frequently estimated using ordinary least squares regression despite the bounded nature of these outcomes. When methods appropriate for bounded outcomes are used, the standard approach is a quasi-likelihood generalized linear model (GLM) with heteroskedasticity- and autocorrelation-consistent (HAC) standard errors. However, no existing estimator jointly models the beta-distributed conditional density and autoregressive (AR) error structure. We introduce betark, a Stata implementation of a joint conditional maximum likelihood estimator for beta regression with AR(k) errors based on a recursive substitution that yields a closed-form conditional beta likelihood with autoregressive dependence of arbitrary order. Unlike two-stage approaches, betark jointly estimates the mean, precision, and AR(k) coefficients in a single likelihood, so reported standard errors directly account for autocorrelation without separate correction. A Monte Carlo study compared betark with a quasi-binomial GLM using Newey-West HAC standard errors across AR(1)-AR(3) processes, three series lengths, and four effect sizes in a single-group ITSA design. Both methods were essentially unbiased, but betark produced better-calibrated inference than GLM+HAC in most scenarios, with the largest gains under highly persistent autocorrelation, where GLM+HAC Type I error exceeded 60% for short series. Misspecifying the AR order by one lag and varying the starting mean and pre-intervention trend had only modest effects on performance. However, betark's own Type I error remained elevated under highly persistent AR(3) processes even for the longest series examined.

The logistic-normal integral appears in problems of statistical estimation for logistic models with Gaussian random effects, and generalized linear mixed models. We study the numerical evaluation of this integral and of its derivatives, and give closed form evaluations at certain points and series expansions. There is a continuum of possible series expansions, and we single out one series expansion which is optimal for numerical evaluation. We propose an algorithm for a precise numerical evaluation, based on the optimal series, with good approximation error control in the tails. As an application we give explicit results for the first four moments of a logistic-normal random variable.