Papers

Randomized saturation designs are widely used to study spillover effects in clustered populations. In these designs, clusters are first assigned to treatment saturation levels, and units are then randomized within clusters according to the assigned saturation. This paper develops randomization tests for such experiments under several null hypotheses that arise naturally in spillover analysis. For a fixed pair of saturation levels, we first study two individual-level hypotheses: a partially sharp null of no spillover effect for every untreated unit and a bounded null that restricts individual spillover effects by a prespecified constant. Both hypotheses can be tested using a common conditional randomization framework, with finite-sample validity obtained by combining the same focal-unit relabeling distribution with null-specific statistics. We then study weak average-spillover nulls and show that, although these nulls do not yield finite-sample exact conditional tests, studentized relabeling statistics deliver asymptotically valid randomization-based inference. Finally, for multiple ordered saturation levels, we develop a finite-sample valid unconditional pairwise-imputation test for global monotonicity of spillover effects. Simulations and an application to the Zomba Cash Transfer experiment illustrate the finite-sample behavior and practical implementation of the methods.

Bayes-assisted conformal prediction combines the strengths of Bayesian modelling with exact, distribution-free frequentist coverage guarantees. Although conformal validity is preserved even when the Bayesian working model (BWM) is misspecified, the size of the resulting prediction sets can degrade substantially when the prior is poorly aligned with the observed data. We address this limitation by introducing RoBAS (Robust Bayes-Assisted Shrinkage): a Bayes-assisted framework for constructing robust nonconformity scores, with two instantiations: one induced by a heavy-tailed BWM, and a closed-form empirical Bayes shrinkage score. The resulting scores adapt to the quality of the working information encoded in the prior: when this information is reliable, they exploit it to produce efficient prediction sets; when it is weak or inaccurate, they revert to the Distance-To-Average (DTA) score, a robust non-informative baseline. We evaluate the proposed scores on tabular and image regression tasks where the training distribution may differ from the calibration and test distributions, while the calibration and test data themselves remain exchangeable. We find that they are competitive with widely used scores in the absence of such shift, while substantially reducing interval widths in shifted settings.

Statistical modeling of rainfall amounts is of considerable importance in meteorology, hydrology, and agriculture. The gamma distribution remains one of the most popular choices for modeling rainfall data due to its flexibility and ability to capture the skewness of rainfall observations. Rainfall datasets often contain atypical observations due to measurement errors and extreme weather events, making maximum likelihood estimation (MLE) highly sensitive to contamination. In this paper, we develop a robust estimation framework for the two-parameter gamma distribution based on the minimum density power divergence estimator (MDPDE). Explicit estimating equations are derived, and several theoretical properties of the proposed estimators are established. In particular, closed-form expressions for the asymptotic covariance matrix are obtained, and robustness is investigated through influence function analysis and asymptotic relative efficiency. The finite-sample performance of the estimators is examined through simulation studies under both pure and contaminated gamma models. The proposed methodology is further implemented to analyze detrended areally weighted monsoon rainfall data from the 36 meteorological subdivisions of India for the period 1951–2014. The results demonstrate that the MDPDE provides a useful compromise between robustness and efficiency, yielding more stable inference than MLE in the presence of outliers while maintaining high efficiency for uncontaminated data.

Confirmatory phase III trials require rigorous evidence, yet for first-in-class (FIC) therapies they must often be designed when same-mechanism evidence is scarce. This uncertainty motivates planned interim analyses and makes phase II data from the same therapy a relevant source of prior evidence. However, both borrowing and repeated interim analyses must be calibrated to control the overall type I error rate. Because borrowing changes the evidence available at interim analyses relative to a non-borrowing group sequential design (GSD), it also raises the question of whether interim analysis timing should be prospectively adapted to the borrowing-adjusted evidence base. We propose a prespecified adaptive interim-timing framework based on Bayesian information borrowing and Bayesian predictive probability $B^2$-FIC. The borrowing model is calibrated against phase II–phase III discrepancy scenarios to control overall type I error rate. At the first interim analysis (IA1), the calibrated model combines phase II information with accumulating phase III data to update the posterior. Bayesian predictive probabilities from this posterior select the earliest information fraction for the second interim analysis (IA2) that meets the efficacy criterion. In simulations, $B^2$-FIC maintained empirical type I error and improved interim power across different scenarios. Predictive probabilities derived from phase II and phase III IA1 data selected earlier IA2 than GSD when evidence was favorable. Two oncology case studies illustrate the framework. Overall, $B^2$-FIC provides a calibrated framework for adapting interim timing to borrowing-adjusted evidence, an emerging design problem in confirmatory trials.

Emerging climate risks pose pressing challenges for insurers, governments, and businesses worldwide, as they face growing uncertainty in quantifying the impacts of climate change from physical risks. This paper aims to understand the impact of specific climate factors on mortality by cause for subgroups of the United States (US) population, through sensitivity and scenario analysis based on an increase in temperature and sea level extremes. We apply compositional data analysis (CODA) techniques to examine cause-specific deaths, treating the density of deaths as a set of dependent, non-negative values that sum to one. We couple CODA with principal component analysis on the climate factors as a means of dimension reduction, and fit generalised additive models to better reflect the non-linear relationships between the dimension-reduced principal scores and mortality by cause. The results of our analysis indicate climate-related factors have varying impacts by cause and ages within each cause, with more pronounced increases in the proportions of deaths from hypertensive heart disease as temperature and sea level extremes increase. Scenario analysis also indicates that an increase in temperature high extremes, sea level, and rainfall, in conjunction with a decrease in low temperature extremes, lead to offsetting impacts on the proportion of deaths between climate-related causes, but less offsetting by age within causes. Furthermore, the impacts on the proportions of death are more pronounced for ages between 55 and 95, reinforcing the observation that climate-related risks have a greater impact on older (and potentially more vulnerable) subgroups of the population. For life insurers specifically, these results are consistent with the natural hedge that arises between annuity and protection products, in light of increasing climate risk.

In-context learning (ICL) has emerged as a central capability of pretrained language models, yet its theoretical analysis has focused primarily on causal language models trained by left-to-right autoregressive prediction, such as GPT-style models. Masked language models instead recover masked tokens from bidirectional context, and their role in ICL remains less understood. We develop a statistical learning framework that represents the context examples by their empirical measure and models prediction as a function of the context and the query. This formulation places autoregressive and masked pretraining objectives within a common excess-risk analysis. Under Wasserstein-type regularity conditions, we relate pretraining with T tasks and N samples per task to k-shot excess risk at inference, obtaining same-order upper bounds for masked and autoregressive objectives. We also study task-distribution shift, where pretraining tasks are sampled from P and inference tasks from Q; the resulting bound contains an additional term controlled by the lifted Wasserstein distance between P and Q. The bounds further imply an order-optimal allocation under a fixed pretraining data budget and refined rates under intrinsic low-dimensional structure. Experiments on controlled function-learning tasks show that the Masked Pair Encoder (MPE) can achieve performance comparable to GPT-2-style causal Transformers, suggesting that ICL behavior is not specific to causal language models.

In linear regression, the signs of coefficients convey the direction of covariate effects and are central to empirical interpretation. In high-dimensional settings, however, the abundance of candidate covariates introduces substantial model selection uncertainty. We study the deliberate manipulation of coefficient signs through the inclusion of a carefully chosen auxiliary variable, a practice we term SHAVE (Sign Hacking with Auxiliary Variable Exploration). We show that, conditional on the outcome and variables of interest, there exists a set of auxiliary-variable realizations with positive Lebesgue measure that lead to sign reversals upon inclusion. Moreover, with high probability, such variables can be found when many auxiliary candidates are available, leading simultaneously to reversed signs, inflated $t$- and $F$-statistics. Simulation studies and an empirical application corroborate these theoretical findings. We further propose detection strategies for SHAVE when augmented or independent datasets are available, as SHAVE has important implications for reproducibility, $p$-hacking, and research integrity.

DCEDesignSA is a freely available MATLAB package for generating Bayesian D-optimal discrete choice experiment designs. It employs Simulated Annealing to efficiently search the design space and maximise the Bayesian D-optimality criterion under user-specified prior distributions. The toolbox features an interactive graphical user interface, enabling researchers without programming expertise to define experimental settings, generate optimal designs, and export survey-ready designs directly to Qualtrics. DCEDesignSA supports interaction terms in utility, no-choice alternatives, and presentation order effects.

Distinguishing Large Language Model (LLM) generated text from human writing is a critical and difficult challenge. While LLMs are trained to write like humans, we hypothesize that this training leaves an indelible mark. LLMs develop a particularly strong aversion to token repetition very early in training. This bias persists as a ”Vestigial Heuristic” (a developmental artifact) that is activated in LLM-generated text, separating LLM from human writing. To probe this phenomenon, we introduce Telescope Perplexity, a metric that evaluates the token repetition of the model, $P(s_i | s_{1:i})$ . Our empirical investigation reveals that the Telescope Perplexity signature emerges early in pre-training, and Telescope Perplexity empirically enables highly effective zero-shot LLM detection. We show state-of-the-art or competitive performance across diverse datasets (including modern evaluation sets we introduce), reference models, and perturbation schemes with greater efficiency than other methods.

We study the problem of maximum likelihood estimation for moving average (MA) time series models from the perspective of algebraic statistics, with a focus on the structure and number of solutions of their likelihood equations. Of particular interest is to classify the critical points that lead to non-invertible models. We consider the composite likelihood as an alternative estimation method and analyze its critical points. We extend our algebraic analysis to autoregressive processes (AR). We provide algebraic closed form formulas for the parameters when possible. We also explore in simulations how methods from numerical algebraic geometry perform against traditional optimization for these models.

Joint Embedding Predictive Architectures (JEPAs) have emerged as a promising framework for self-supervised representation learning by predicting latent embeddings of masked regions rather than reconstructing pixels. Existing JEPA methods typically employ a single student encoder, leaving the role of Siamese student encoders largely unexplored. In this paper, we propose Siamese JEPA (SiamJEPA), a JEPA framework with masked Siamese student encoders and an exponential moving average (EMA) teacher, which can also be viewed as a JEPA formulation of the brain-inspired representation learning model PhiNet. We further introduce Random Shuffle Teacher (RST), which removes spatial correspondence in teacher targets to encourage semantic patch representations, and develop an RST-based semantic-to-spatial curriculum. Experiments on ImageNet show that Siamese student encoders effectively regularize the JEPA objective, improving representation separability and accelerating early-stage learning. Moreover, under RST, stronger Siamese regularization substantially increases class-discriminative information in individual patch tokens, suggesting that the semantic bias arises from the interaction between RST and the Siamese objective rather than from shuffling alone. SiamJEPA consistently outperforms comparable single-encoder JEPA variants under limited training budgets. With RST-based curriculum learning and a ViT-Base backbone, SiamJEPA achieves 74.2% linear-probing accuracy after 450 epochs using a substantially simpler masking strategy, compared with I-JEPA (72.9%) and DSeq-JEPA (73.5%) after 600 epochs. These results demonstrate that Siamese student encoders provide an effective inductive bias for predictive representation learning and can be further enhanced through semantic-to-spatial curriculum learning. The source code is publicly available at https://github.com/oist/SiamJEPA.

Randomized controlled trials (RCTs) identify treatment effects in the randomized trial population but are often too small for reliable heterogeneity estimation; observational studies (OS) are larger but confounded and measured on only partially overlapping covariates. We develop Bayesian Calibrated ALignment under covariate Mismatch (B-CALM), a Bayesian borrowing method for RCT-defined conditional average treatment effect (CATE) estimation. B-CALM maps source-specific covariates into a shared latent state, jointly models trial and observational outcome surfaces, and uses baseline-bias and comparative-bias functions to represent how the OS departs from the trial estimand. The comparative-bias prior becomes an explicit sensitivity knob: we prove a finite-feature bias-limited information bound showing that observational contrast information about the trial treatment-effect function is capped by the prior precision of this bias function, and derive an effective-sample-size formula showing that the RCT-equivalent information contributed by the OS saturates as OS sample size grows. The theory also combines a PAC-Bayes trial-risk bound with an integral-probability-metric (IPM) alignment and calibration decomposition that separates RCT empirical risk, latent alignment, and residual calibration of the debiased OS surface. In synthetic, semi-synthetic, and pediatric-obesity external-control studies, B-CALM maintains near-nominal average coverage and low negative transfer while pooled and causal-forest baselines can become overconfident under comparative bias.

Differential privacy is a mathematical definition of what it means to protect data subjects' privacy in data releases. Differential privacy depends on a parameter $\epsilon$ known as the privacy budget. The value of $\varepsilon$ determines the nature of the privacy guarantee, with smaller values generally offering more privacy. However, reducing $\varepsilon$ also tends to decrease the accuracy of results protected with differentially private algorithms. Setting a value for $\varepsilon$ that satisfactorily balances this risk/accuracy trade off is complicated in practice, and there is not a standard approach to doing so. In part this is because practitioners may struggle to understand the privacy guarantee afforded by $\varepsilon$. We present an approach to interpreting and setting $\varepsilon$ in which (i) the practitioner establishes bounds on the posterior odds that adversaries can learn sensitive information, and (ii) the practitioner converts these bounds to values of $\varepsilon$. We illustrate the approach using data from a case control study.

Estimating heterogeneous treatment effects (CATE) requires simultaneously detecting effect modification and quantifying estimation uncertainty. Existing tree-based methods make an uneasy trade-off: significance-based approaches (Radcliffe and Surry 2011) identify subgroup interactions directly but lack valid inference; honest causal trees (Athey and Imbens 2016) deliver nominal confidence interval coverage but use outcome-agnostic splitting criteria that sacrifice interaction sensitivity. We introduce a hybrid algorithm that fuses significance-based splitting with honest sample-splitting and cross-validation. Our splitting criterion uses the squared $t$-statistic for the treatment $\times$ side interaction ($t^2$), which is shown to be directly aligned with the honest $\text{EMSE}_\tau$ criterion when the interaction is strong. Post-hoc honest cross-validation selects the cost-complexity penalty, giving a single principled estimator with nominal CI coverage at the leaf level. For forests, we retain bootstrap count vectors to enable an infinitesimal jackknife (IJ) variance estimate of Monte-Carlo convergence rather than formal pointwise inference. On the three synthetic designs from (Athey and Imbens 2016) the single tree achieves approximately 90% leaf-average CI coverage at the 90% nominal level across all three designs (200 replications each); on the Criteo, Hillstrom and Starbucks uplift datasets we match Qini coefficient performance of S-, T-learner and GRF baselines. An open-source Python package with reproducible seeds, sklearn-compatible API, and full test coverage accompanies this work (https://codeberg.org/hadjipantelis/rattus).

In a 1934 Annals of Mathematics paper, Samuel S. Wilks posed the problem of computing joint moments of disjoint principal minors of Wishart random matrices, describing the general case as “extremely complicated.” These moments arise in classical multivariate statistics, including multivariate regression, analysis of variance, generalized correlation coefficients, and likelihood ratio testing. We solve Wilks' problem for nonnegative integer exponents and any finite number of disjoint principal minors using Stein's method, specifically the Wishart Stein characterization recently developed by Bailly et al. (2026). The Wishart Stein identity yields a degree-lowering recursion that terminates after finitely many steps and evaluates the desired moment exactly. Numerical experiments validate our recursive formulas against direct Monte Carlo simulations from the Wishart distribution.

Recent work in predictive Bayesian inference has enabled novel Bayesian interpretations of many well-known stochastic one-step-ahead predictive algorithms. In this paper, we study classic kernel density estimation in the predictive Bayesian framework. We prove that their predictive measures converge weakly almost surely $\unicode{x2013}$ meaning that their associated predictive resampling sequences are almost surely asymptotically exchangeable $\unicode{x2013}$ and we provide estimators for moments of the limiting random probability measure. We also show that the resampling sequences do not satisfy standard assumptions like being conditionally identically distributed (c.i.d.) or almost c.i.d. (a.c.i.d.), thus providing a non-trivial example of a predictive sequence which is not a.c.i.d. but nevertheless converges weakly almost surely. For Gaussian kernels, we show that the limiting directing measure is almost surely absolutely continuous with respect to the Lebesgue measure, meaning it emits a probability density. This enables us to derive credibility intervals for kernel density estimates, which we illustrate on two real datasets.

Local Gaussian correlation (LGC) measures dependence locally, making it a natural tool for tail dependence and financial contagion, but its estimates degrade in the joint tails, where they are most needed. Location-adaptive bandwidths have been tried for LGC and found inferior to a single global bandwidth; we explain why, and map the regime in which adaptivity does help. First, a diagnostic: across heavy-tailed data-generating processes the parametric marginal pre-transform is inert (it changes the integrated error only in the fourth decimal), while the binding constraint is the local effective sample size, with the replication dispersion following a Fisher variance floor sd ~ (1 - rho^2)/sqrt(eff_n). Second, theory: specializing the Hjort-Jones local-likelihood asymptotics to the bivariate Gaussian family that LGC fits, we derive the first location-specific AMISE-optimal bandwidth for LGC, b*(x) proportional to [(1 - rho^2)^2 / (f beta^2)]^(1/6) n^(-1/6), and validate its bias expansion directly (bias proportional to b^2 beta, R^2 approximately 0.9, slope-to-beta correlation 0.80). Third, a regime map: a Monte Carlo across dependence strengths shows the adaptive rule beats the global plug-in only at moderate dependence with curved surfaces. At weak dependence there is no curvature to exploit; at strong dependence finite-sample bias from the steep surface dominates, and adaptivity performs substantially worse, with an error that grows in the sample size. This explains the field's experience that global bandwidths are hard to beat, and locates the exception. Fourth, application: on volatility-filtered equity returns the adaptive estimator yields more stable tail-dependence surfaces under resampling. The message is cautionary: the binding constraint on tail LGC is data scarcity, not bandwidth placement, and no bandwidth, however optimal, can recover information the data do not contain.

Minimum maximum mean discrepancy (MMD) estimation has emerged as a robust and likelihood-free alternative to maximum likelihood estimation for parameter estimation. Yet, despite its practical success, the associated optimization problem remains poorly understood, with theoretical guarantees for existing algorithms hinging on convexity assumptions that rarely hold in practice. We address this gap by proposing a preconditioned gradient descent (PGD) scheme, establishing its asymptotic global convergence under explicit gradient-dominance and projection-residual conditions. Our approach is inspired by recent progress on MMD gradient flows, a nonparametric descent scheme on the space of probability measures. We provide extensive empirical evidence that our PGD scheme outperforms standard gradient descent across a range of challenging parameter estimation and composite hypothesis testing problems.

The increasing granularity of third-party weather and exposure information can allow insurers to more effectively predict weather-related losses. However, loss outcomes are often reported in spatially grouped observations, such as at the county level, so higher resolution predictors are aggregated to align with the granularity of the outcome in standard analyses. Assuming an underlying zero-inflated mixed-effects spatial point process framework for claims arising from a common storm, we derive a model for unbalanced, multivariate zero-inflated count data that incorporates rich weather and exposure predictors observed at higher spatial granularity to predict claim patterns. The model accommodates the dependence between locations affected by a common storm in the excess zeros, as well as in the joint claim counts. Using real property exposure and loss data, we emphasize the value of incorporating granular predictors to address the localized heterogeneity of storm losses.

{kindling} is an R package that provides a higher-level interface to {torch}, R's native implementation of PyTorch, for defining, training, and tuning neural networks. It supports multilayer perceptrons and recurrent architectures (RNN, LSTM, GRU) while reducing the boilerplate typically required to write {torch} model definitions and training loops by hand. The package is organized around three levels of abstraction: code-generation functions that return inspectable, unevaluated {torch::nn\_module()} expressions; direct-training functions that fit a model from a formula and data frame; and {tidymodels}-registered model specifications that let neural networks be fit, tuned, and evaluated using the {tune}, {dials}, {recipes}, and {workflows} infrastructure that {tidymodels} users already rely on for other model types. This design lets analysts move from exploratory training to systematic hyperparameter search without leaving the {tidymodels} ecosystem, while retaining the ability to inspect or modify the generated model code rather than treating it as a black box.

We propose a Targeted Highly Adaptive Lasso for estimation of non-pathwise differentiable functional parameters such as the dose-response curve (DRC) for continuous exposure. We assume the target function lies in the $k$-th order smoothness class used to define the $k$-th order Highly Adaptive Lasso (HAL), which can be well approximated by linear spans of $k$-th order spline basis functions. We construct a projection of the true target function onto a large finite dimensional working model spanned by an initial set of $k$-th order spline basis functions, which defines a pathwise differentiable approximation of the target functional parameter. A standard TMLE is then applied with a data-adaptive initial fit, replacing the MLE targeting step with a LASSO step over HAL spline basis functions that span the target function. We prove that the resulting Targeted HAL-MLE is pointwise asymptotically normally distributed and achieves a convergence rate determined solely by the dimension and smoothness of the target function, giving dimension free rates up till $\log n$-factors. Through a simulation study for the DRC, we show that the Targeted HAL outperforms a HAL plug-in estimator in terms of bias and mean squared error. Targeted HAL offers a fully data-adaptive approach to inference on functional parameters without requiring sieve specification or parametric assumptions.

Normalising flows provide a powerful variational family for approximate inference, yet individual architectures often fail to generalise across heterogeneous posterior geometries. We revisit mixture-based flow formulations and introduce \emph{AMF\mbox{-}VI\mbox{-}sEMA}, a two-stage framework featuring a stable global weighting mechanism based on a Simplex Exponential Moving Average (sEMA) update. In Stage~1, a heterogeneous set of experts (RealNVP, MAF, RBIG) are trained independently to specialise in distinct structural regimes. In Stage~2, expert parameters are frozen and global mixture weights are learned through a temperature-controlled softmax of average log-likelihoods, followed by a smooth EMA update on the probability simplex. This design produces a tractable, data-agnostic gating mechanism (without per-sample gating or gradient backpropagation through weights) that adaptively reallocates capacity while avoiding component collapse. We evaluate the framework on ten posterior benchmarks: six canonical 2D synthetic families (Banana, X-Shaped, Bimodal, Multimodal, Two-moons, Rings) and four real/low-dimensional Bayesian targets (BLR, BPR, Weibull, Real-GMM2), with stronger baselines (NICE, ResFlow, and EM-Mixing). Comprehensive evaluation covers NLL, KL divergence, Wasserstein-2 distance, and MMD, together with diagnostics of mixture dynamics, hyperparameter sensitivity, and cross-seed robustness. Empirically, \emph{AMF\mbox{-}VI\mbox{-}sEMA} achieves consistent NLL improvements over its predecessor \emph{AMF\mbox{-}VI} and avoids the catastrophic transport failures of single-flow baselines, while maintaining stable weight trajectories ($N_{\mathrm{eff}}{>}1.4$ on all datasets) with minimal computational overhead.

In this article, we investigate the asymptotic Bayes optimality under sparsity (ABOS) of the Gavrilov-Benjamini-Sarkar (GBS) step-down multiple testing procedure of Gavrilov et al. (2009) in the sparse Gaussian sequence model. While the asymptotic optimality properties of the Benjamini-Hochberg procedure have been extensively studied, corresponding results for the GBS procedure remain unavailable despite its favorable finite-sample performance and widespread applicability. Within the spike-and-slab Bayesian formulation and the asymptotic decision-theoretic framework of Bogdan et al. (2011), we establish that the GBS procedure is ABOS over a broad class of sparse asymptotic regimes. Existing ABOS analyses of the Benjamini-Hochberg procedure rely on approximating the random rejection threshold by a suitable deterministic surrogate. In contrast, our approach analyzes the Bayes risk directly, separately controlling the false discovery and false nondiscovery components. The analysis combines two key ingredients: a new finite-sample inequality for shifted order statistics associated with the GBS critical constants and a signal-crossing argument for ordered alternative p-values that exploits the procedure's sequential step-down structure. Together, these tools yield the desired ABOS property without resorting to threshold-localization arguments. To the best of our knowledge, this is the first asymptotic decision-theoretic analysis of the GBS procedure and the first proof of its asymptotic Bayes optimality under sparsity.

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=\Lambda(x)U(x)$. The diagonal factor $\Lambda(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

Bootstrap-based methods have been recommended for assessing prediction instability in clinical prediction models, but their performance relative to cross-validation (CV) remains unclear. We propose a CV-based approach for assessing prediction instability and compare it with a bootstrap-based approach in logistic regression and random forest models. We conducted a resampling-based empirical experiment using a clinical cohort of 19,418 emergency department patients. Development samples were generated under events-per-variable (EPV) scenarios of 10, 30, and 50, and results were compared with those from the full dataset. Models were evaluated using bootstrap validation and repeated 5-fold CV; nested CV was used for random forest tuning. Predictive performance was assessed using AUC, calibration slope, and calibration-in-the-large. Prediction instability was quantified using mean absolute prediction error (MAPE). For logistic regression, bootstrap validation and repeated 5-fold CV produced broadly similar discrimination and calibration, especially at higher EPV values. For random forest, apparent performance consistently overestimated empirical discrimination. Bootstrap validation and repeated 5-fold CV gave comparable discrimination, but repeated 5-fold CV produced calibration slope estimates closer to the empirical value. Prediction stability improved as EPV increased for both modelling approaches. At EPV 30, bootstrap-derived MAPE was higher than CV-derived MAPE for both logistic regression (median, 0.042 versus 0.020) and random forest (median, 0.077 versus 0.027). A CV-based approach can assess prediction instability while also providing internally validated performance. These findings support CV-based instability assessment as a practical alternative to bootstrap-based assessment, particularly when comparing instability across multiple modelling algorithms.

The Gaussian Error Linear Unit is usually motivated as the expected output of an input-dependent Bernoulli gate. This work gives an alternative interpretation: GELU is the expected output of a hard linear gate with a Gaussian random threshold. This view provides a generative interpretation for the Bernoulli gate: the gate opens once the input clears a latent Gaussian threshold. This interpretation stems from a decomposition based on well-known results in stochastic inventory theory and leads to a threshold-transmission family that includes ReLU, GELU, SiLU/Swish, and hard swish as special cases. By considering a latent uniform threshold, we recover a hard-swish-like piecewise-polynomial gate whose nonlinear transition is confined to a finite interval, yielding fixed- and learned-width variants. Controlled experiments on compact vision and language models show that calibrated or learned uniform-threshold gates are consistently competitive with GELU, ReLU, and SiLU/Swish, display architecture-dependent learned widths, and use the finite transition region nontrivially.

Modern sequence models have a striking capacity for in-context learning (ICL); they can perform new tasks based only on examples given in the prompt. Understanding how this ability emerges requires theory that captures important properties of natural data. Linear regression has served as a useful sandbox for ICL theory, but existing work has largely focused on prompts with independent examples. In this work, we extend this setting to sequentially correlated data, a basic feature of real sequences. We present a solvable model based on linear attention and test our predictions on realistic transformer architectures. We identify two distinct effects: First, when the query token is independent of the context, within-context correlations induce an effective context length: correlated prompts behave like shorter i.i.d. prompts. Second, when the query is also correlated with its context, test error is reduced, particularly for softmax attention when compared to linear attention. These results suggest that correlated prompts alter not only the effective sample size of in-context learning, but also which attention architectures are best matched to the task.

The manifold hypothesis posits that high-dimensional data are concentrated near a low-dimensional embedded manifold. Recent advances in mixture variational autoencoders (VAEs) provide a powerful tool for extracting such underlying structure in a faithful manner. The resulting geometric structure naturally introduces local and global relationships among variables, thereby providing a systematic way of imputing missing data. We propose a model-based imputation method that enables sampling from \( p(\bm{x}_{\mathrm{mis}} \mid \bm{x}_{\mathrm{obs}}) \) via a sampling-importance-resampling (SIR) procedure, which can be further augmented with a joint diffusion model in the latent space. Our method imputes missing data while respecting the underlying geometry, achieves competitive performance compared to state-of-the-art procedures, quantifies uncertainty in the imputations, and is model-based, thereby enabling on-the-fly imputation without rerunning the entire procedure.

For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison–Reiman data with a nonsingular $M$-matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular $M$-matrix assumption is structural. In the larger completely-$\mathcal{S}$ class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai–Dieker question has a positive answer in the Harrison–Reiman $M$-matrix class and a negative answer in a natural completely-$\mathcal{S}$ extension.

Recent advances in generative modeling have enabled the efficient computation of Schrödinger bridges (SB) in high-dimensional settings by leveraging partially simulation-free training methods inspired by flow matching. However, these have not covered SBs with reflecting dynamics, a useful model choice with built-in guarantees that generated samples stay in the data domain. Existing alternatives for reflected SBs instead rely on more complex training based on forward–backward SDE theory, requiring expensive higher-order derivatives and sampling entire paths during training. In this article, we introduce a partially simulation-free framework that allows reflected SBs to be trained similarly to flow matching, using a new sampling method and regression target. We demonstrate our results by coupling pairs of well-known high-dimensional image datasets. Using reflected dynamics incurs negligible additional wall-clock time during both training and inference while maintaining or slightly improving generative performance.