Papers

Behavioural randomized controlled trials (RCTs) are the empirical workhorse of behavioural science, yet quantum computing has reached the field only through opinion-dynamics demonstrations and decision-theory formalism, leaving the RCT analysis pipeline unexamined. I ask, stage by stage, where quantum or quantum-like computation earns its place against a strong classical baseline, and report three controlled simulation pilots across the pipeline. In a power-calculation pilot anchored on a 97-outcome preprocessing multiverse of real behavioural RCTs, quantum amplitude estimation recovers enumerated multiverse-significance fractions with absolute error 20-30 times smaller than classical sampled Monte Carlo at matched query budgets on the two non-degenerate outcomes, conditional on efficient state preparation. In an interference pilot, an entanglement-structured estimator recovers a four-way spillover that a pairwise model cannot represent at any sample size. In an adaptation pilot, a quantum-inspired contextual bandit is decisively beaten by a correctly specified classical baseline. The three results - one conditional win, one representational win, one honest loss - yield a decision rule for when quantum methods are worth adopting, piloting, or deferring, together with a five-gap research agenda.

Sparse longitudinal CT follow-up limits lesion-size forecasting when only a few prior observations are available. We constructed a five-visit DLT-derived same-lesion trajectory benchmark from DeepLesion and Deep Lesion Tracker (DLT), yielding 205 trajectories from 129 patients. We compared an exploratory conventional sparse-to-final analysis with a primary fixed visit-index horizon design predicting the common log change from T3 to T4 while progressively adding earlier observations, evaluating predictive accuracy, uncertainty reliability, post-hoc conformal interval calibration, subgroup performance, and Gompertz-inspired trajectory regularization. The evaluated methods showed partially overlapping point-prediction accuracy but distinct uncertainty behavior. Mean held-out RMSE across ten training seeds was 0.4726, 0.4305, 0.4499, and 0.4513 for m = 1, 2, 3, 4, indicating the lowest mean RMSE at m = 2; additional history did not improve RMSE. At m = 4, raw Cohort-Level Feature GP coverage was near the 95% nominal level, whereas MC Dropout, Deep Ensemble, and residual-scale intervals were conservative. Patient-level conformal calibration generally produced near-nominal or conservative coverage at the cost of wider intervals. Patient-grouped development cross-validation selected lambda* = 0 for the Gompertz-inspired term. A global population reference frequently opposed lesion-level change directions, and prediction difficulty varied across anatomical subgroups. Overall, additional historical observations provided limited predictive benefit once the prediction horizon was controlled, while predictive accuracy, uncertainty reliability, and trajectory consistency did not necessarily improve together, and should be evaluated jointly in sparse longitudinal imaging.

Accurate rare-event estimation can be computationally expensive. Classical adaptive importance sampling (IS) schemes often rely on restrictive proposal families and can struggle under multiple failure modes. We propose FAMIS, a flow-based multiple importance sampling (MIS) framework that learns a nonuniform mixture of normalizing flow proposals for rare event estimation. The method does not require presampled failure data or prior knowledge of the number, location, or geometry of the failure modes. Instead, it adaptively learns the mixture through sequential evaluations of the limit state function. To guide training toward the failure domain, FAMIS uses a smooth rare-event surrogate and a tempered target sequence. A defensive exploration mixture improves early-stage coverage, a Rao Blackwellized update adapts the mixture weights, and a Jensen-Shannon repulsion term promotes separation and diversity among the base components. The final failure probability is computed with a deterministic-mixture MIS estimator. Numerical experiments demonstrate that FAMIS accurately approximates quasi-optimal IS densities with fewer training samples and model evaluations, providing stable variance reduction across complex reliability problems.

We consider the identification of memory and stochastic forcing in a one-dimensional generalized Langevin equation with state-dependent Brownian noise and optional additive symmetric $\alpha$-stable jumps. A finite-dimensional Markov embedding provides Kramers–Moyal conditional moment relations that separate memory drift, Brownian diffusion, and jump contributions. From increment ensembles conditioned on the observed velocity history, truncated statistics estimate the stable parameters, corrected second moments determine the Brownian amplitude, and first moments lead to a Volterra equation for the memory kernel, solved by RKHS regularization. We establish conditional finite-sample bounds for the moment estimators and a conditional $L^2$ error bound for kernel recovery with a data-selected regularization parameter. Numerical experiments examine Brownian and Brownian–Lévy models under different forces, memory kernels, observation lengths, measurement noise, and approximate history conditioning.

Sampling many responses and keeping one that passes a verifier lets large language models solve problems beyond their single-response ability, but this search must be paid again for every query, while many deployments answer with a single response. Post-training on verified responses can transfer the benefit of search into the model. With a fixed budget, selecting by correctness alone spends slots on problems the model already answers correctly, leaving fewer to correct its failures. To address this imbalance, we propose GapFT, which trains on the gap between Pass@K and Pass@1: problems that the source model fails with one response but solves within K samples. GapFT keeps the objective and training budget fixed and changes only which verified responses enter training; an exact decomposition splits the resulting Pass@1 change into corrected failures and regressions on problems the source model already solved. On LogiQA 2.0 and ReClor with three model families, GapFT is above budget-matched uniform rejection-sampling fine-tuning (RFT) in every setting, with a positive pooled effect, and on Llama-3.1-8B and Mistral-7B it recovers about two thirds to four fifths of the gain of fine-tuning on the entire verified pool with 11-34% of its problems. Further analyses reveal that the gain comes from failures that the first few search samples recover, while failures found only by deeper search displace replay and add no net gain, that filling the same budget with gold-labeled failures search cannot reach lowers accuracy, and that the gain is bounded by how many transferable failures search exposes.

Industrial predictive maintenance increasingly depends on learning from equipment spread across sites whose sensor data cannot easily be pooled. Federated averaging (FedAvg) solves this with a central aggregation server; gossip learning removes the server, but its behaviour for recurrent failure-detection models has not been measured under controlled conditions. We compare synchronous ring gossip with FedAvg, isolated local training and a centralized reference for a stacked LSTM that detects imminent failure on the NASA C-MAPSS turbofan benchmark. All methods share one open implementation, architecture, initialization, optimizer, data split and training budget, and the primary endpoint uses one terminal window per test engine to avoid the statistical dependence of overlapping windows. On FD001 (five seeds), gossip reached a terminal-window F1 of 89.6 +/- 1.3%, compared with 89.9 +/- 1.1% for FedAvg, 83.6 +/- 6.7% for local training and 93.5 +/- 2.1% for centralized training, while transmitting the same payload as FedAvg without a coordinator. Node models agreed closely but not exactly (1.8% pairwise decision disagreement versus 5.6% without communication). Across FD002-FD004, peer communication improved terminal-window F1 over local training by 13-28 points; gossip matched FedAvg on FD003 and FD004 but was 4.3 points lower on the multi-condition FD002 subset. Simulated message loss, node failure and server outage changed neither method appreciably, whereas larger rings degraded gossip faster. Ring gossip is therefore a practical serverless alternative when data heterogeneity is moderate, and faster-mixing topologies become important as heterogeneity grows.

This paper studies a two-server split–merge queue with positively dependent subtask service times modeled through a latent-factor bivariate Erlang construction. An exact characterization of the split–merge completion time is obtained, including explicit formulas for its first two moments and the resulting mean waiting time. Under fixed marginal service-time distributions, independence is shown to stochastically increase the completion time and hence overestimate mean waiting time. Numerical illustrations show that this benchmark gap can be substantial.

We propose a decoupled, layerwise method for structurally sparsifying the fully connected layers of pretrained neural networks. Rather than penalizing all layers jointly, our approach extracts shallow two-layer subnetworks, normalizes the inner weights, and applies a structured group penalty to the outer weight matrix of each block, processing layers sequentially to prune neurons and reduce the width of each layer. We prove that the constrained decoupled objective is equivalent at optimality to a specific joint penalty on the inner and outer weights, for any positively homogeneous activation, and thus admits a clean projected and proximal formulation. Our central finding is that this decoupled reformulation is more robust than coupled methods. In numerical experiments it provides a wider usable range of the regularization strength and a lower rate of catastrophic over-pruning than the tested joint baseline while maintaining comparable accuracy. We establish these properties in controlled classification and sparse-recovery studies, and examine their scope in a high-dimensional PINN stress test and in the feed-forward layers of OPT-1.3B.

A treatment benefit predictor (TBP) is an algorithm that maps a patient's characteristics to their putative benefit from a given treatment, which can be used to inform treatment decisions. However, a TBP must be evaluated in the target population before being adopted for patient care. When only observational data are available, evaluating TBPs requires standard causal identification assumptions, as treatment assignment in such settings is not random. We obtain the posterior distributions of predictive performance measures to evaluate prespecified TBPs using observational data, by taking advantage of Bayesian additive regression trees (BART). We illustrate the evaluation of TBPs using selected measures and graphical visualizations: the concentration of benefit ($C_b$) index and the moderate calibration curve. Simulation studies of binary and continuous outcomes settings, including balanced and imbalanced treatment allocation in the binary setting, establish the validity of the proposed approach. In a case study, we use this approach to assess a TBP for systemic antibiotic therapy for patients with chronic obstructive pulmonary disease (COPD). We show that the constructed TBP does not in fact make calibrated predictions, because it both makes optimistic predictions of risks and exaggerates the risk reduction due to treatment. We conclude that flexible Bayesian approaches have the potential to assess TBPs, offering opportunities for flexible model specifications, adjustment for confounding, and uncertainty characterization.

Gaussian processes (GPs) provide principled probabilistic predictions while encoding prior knowledge, including equivariances. Yet, their use in large-scale scientific problems is limited by computational cost. Equivariant neural networks are common but typically lack the uncertainty quantification offered by GPs, which is valuable in applications such as molecular research. High-dimensional inputs and large symmetry groups further demand scalability. We establish results pertaining to the interplay of GP equivariance and conditioning and leverage them to obtain equivariant sparse GPs through suitable mean functions and covariance kernels. We instantiate this framework with a flexible class of integration-free equivariant kernels, yielding scalable and data-efficient GP inference. In particular, we introduce triply scalable equivariant Gaussian processes. We employ equivariant sparse variational Gaussian processes for $\mathrm{SO}(2)$-equivariant vector fields and molecular property prediction. Alongside the SVGP, we develop a matrix-free equivariant full-GP implementation that combines an exact Kronecker reduction with preconditioned conjugate-gradient solves, enabling fast and scalable evaluation of the full joint predictive density. We further compare different approaches for selecting inducing points in the equivariant sparse GP models. Our test cases include synthetic $\mathrm{SO}(2)$-equivariant fields as well as the prediction of electric dipole moments of N-methylformamide based on quantum chemistry simulations, achieving accurate, uncertainty-aware predictions at a fraction of the computational cost of classical GP inference.

The Birnbaum-Saunders (BS) distribution has become a widely used model for positive and asymmetric continuous data. Several extensions of the BS distribution have been proposed, including regression formulations that relate its parameters to covariates. In this paper, we introduce a new regression model for the Generalized Birnbaum-Saunders (GBS) distribution, an extension that has received limited attention in the literature despite offering a stronger physical interpretation. The proposed framework follows the structure of Generalized Additive Models for Location, Scale, and Shape (GAMLSS), allowing covariate effects to be incorporated into all three parameters of the distribution, thereby providing greater flexibility for modeling data. In particular, one of the model parameters has a direct interpretation as the median of the response variable, facilitating the interpretation of covariate effects on the conditional median. Parameter estimation is performed via maximum likelihood, and hypothesis tests are proposed based on the Wald test. The proposed regression model is implemented in R through the gamlss package, allowing access to a broad range of tools for model fitting, diagnostics, and assessment. Monte Carlo simulation studies are conducted to investigate the finite-sample performance of the maximum likelihood estimators. The results show that the estimators exhibit desirable properties, with bias decreasing and efficiency increasing as sample size grows. Additionally, the behavior of the Wald test is investigated, showing good performance for moderate to large sample sizes. Finally, two applications to real data illustrate the practical usefulness of the proposed GBS regression model, demonstrating that it is a flexible alternative for modeling positive and asymmetric data compared to the BS model

We characterize, up to universal constants, the total variation distance between finite products of arbitrary probability measures by a simple formula. A theorem of Latala reduces this expression to one scalar equation. As an application, we characterize the sample complexity of equal-prior binary hypothesis testing uniformly in the weak-detection regime, where squared Hellinger distance alone does not determine the answer, and also characterize the TV distance between multinomial distributions.

We study data-driven early stopping for spectral regularisation methods in the classical non-parametric regression setting. Building on the discrepancy principle, we propose a multi-scale stopping rule that applies to general kernel estimators and show that, unlike previous approaches, it achieves full adaptivity over all smoothness levels in the well-specified case. A key contribution of our work is an extension based on random feature approximations, which reduces computational cost on large datasets while preserving minimax-optimal statistical guarantees. Our procedure not only selects an optimal stopping time but also provides a fully data-driven choice of the number of random features needed to achieve optimal rates. Through the established connection between random features and neural networks in the neural tangent kernel regime, our method further yields a principled, data-driven recommendation for the network width. We prove that the resulting simultaneously chosen width and stopping time allow neural networks to attain minimax-optimal learning rates without prior knowledge of smoothness or capacity parameters.

Latent variable generative models are commonly fit using simple priors over latent variables, but draws from these priors often fail to produce realistic data. This failure is due to a mismatch between the prior and the aggregated posterior, the distribution of latent variables induced by the fitted model and the data. This mismatch is often viewed as evidence that the prior is misspecified and should be replaced. Alternatively, in modern generative models, a two-stage strategy is increasingly used where first, the model is fit, and second, the aggregated posterior is estimated (van den Oord et al.,2017; Rombach et al., 2022.). Synthetic data are then obtained by sampling from this aggregated posterior instead of the prior. To check such procedures, we introduce the aggregated posterior predictive check (APPC). Theoretically, we establish sufficient conditions under which the APPC is asymptotically calibrated. For probabilistic principal component analysis, we show that the APPC can remain calibrated under a misspecified latent prior when pervasive factors permit recovery of the signal space. Experiments with variational autoencoders show that aggregated posterior sampling improves generation for heavy-tailed and clustered data relative to Gaussian prior sampling while performing comparably to models with more flexible latent priors.

Complex problem solving (CPS) with large language models (LLMs) is often framed as a matter of stronger reasoning or longer generation. Yet early-step error amplification, prompt brittleness, and failures to revise incorrect commitments are difficult to explain by missing knowledge or expressive capacity alone. This survey interprets CPS as a sequential estimation-and-decision problem over a latent solution state. A controller maintains a belief about an unobserved solution trajectory, updates it as noisy intermediate evidence arrives, and decides whether to commit, verify, branch, roll back, or abstain to minimize expected loss. Reasoning supplies candidate transitions and interpretations, whereas process control shapes and evaluates those proposals and regulates subsequent transitions and observations. Within this framework, we organize existing methods around five components: explicit state representation, transition structuring, validation and constraint enforcement, search and rollback, and uncertainty management. We also interpret evaluation metrics according to the statistical quantities they estimate. The framework further yields a diagnostic hypothesis: interventions should be most effective when they target the error or uncertainty component implicated by an observed failure. We distinguish systematic, stochastic, and irreducible error together with epistemic and aleatoric uncertainty, and call this alignment problem-control fit and its failure control mismatch. For example, additional sampling may reduce sampling variability while leaving a shared systematic error unchanged. This perspective clarifies what current methods estimate and control, what remains uncontrolled, and why reliable validation, targeted recovery, calibrated uncertainty, and matched-budget evaluation are central open problems.

Matrix-valued time series, arising in diverse fields such as economics, neuroscience, and recommender systems, have become increasingly prominent in modern data analysis. Among various modeling frameworks, the matrix-variate CP-factor model represents an important and widely applicable class for capturing low-rank structures in matrix time series. In this paper, we provide a unified treatment for the identification problem of CP factor models using both analytic and algebraic tools. In particular, we characterize the parameter space as the union of two subspaces, one identifiable and the other non-identifiable. We show that existing estimation methods only apply to an open proper subset of the identifiable subspace. In contrast, we propose a quasi-maximum likelihood estimation (QMLE) procedure for CP factor models, which allows consistent estimation on the whole identifiable subspace. Moreover, we show that the estimated loading matrices by QMLE achieve a faster convergence rate compared to existing approaches. A simulation study and a real application are conducted to demonstrate the finite-sample performance of the proposed method.

Quasars are luminous objects in the universe that exhibit stochastic brightness variations encoding information about the supermassive black holes powering them, and modeling these variations from ground-based survey data time series, known as light curves, is a statistical challenge. This paper reviews how stochastic differential equations (SDEs) have been adapted with neural network parameterizations to overcome this challenge in history. We create the Continuous-Delayed-Memory Stochastic Gradient Descent which depend on the past state of the discrete iteration process. We performed the simulation on some 2-dimensional landscape and observed some wider-exploration and more precise convergent behavior compared to Vanilla SGD by adjusting hyperparameters. Besides, we proposed a reinforcement learning structure with continuous time policy gradients for exploratory policies without solving HJB PDE, and we show that its optimality conditions recover the Gibbs policy of previous works.

We study a minimal classification problem: Given independent labeled observations $X\sim P$ and $Z\sim Q$ from two unknown distributions $P,Q$, and given an independent target $Y$ drawn with equal probability from $P$ or $Q$, can one classify $Y$ strictly better than chance whenever $P\neq Q$? The one-nearest-neighbor rule succeeds for every pair of multivariate Gaussian distributions with distinct means and a common positive-definite covariance matrix but can perform strictly worse than chance even for smooth densities on the real line. We construct a fixed randomized kernel rule whose expected accuracy is exactly $1/2+\operatorname{MMD}_k^2(P,Q)/4$, and obtain characteristic kernels on countably generated measurable spaces from countable families of measurable binary questions. We also prove that a deterministic order rule on $\mathbb R$ beats chance for every pair of distinct Borel probability measures. A measurable encoding then gives a deterministic distribution-free rule which beats chance on every countably generated measurable space, in particular every separable metric space. Finally, we show that no rule works for every distinct pair of distributions and every unknown unbalanced class prior; under adaptive target-class selection, every rule other than a fair coin is strictly worse than chance for some finitely supported pair.

We consider the problem of testing Granger causality in time series, specifically, whether the future outcome $Y_{t+1}$ and the exposure history $\bar X_t$ are conditionally independent given the history of $\bar Y_t$ and a set of confounding variables $\bar Z_t$ up to time $t$. This testing procedure distinguishes true causal effects from associations driven by common external processes, supporting reliable decision-makings in applications such as finance, neuroscience, and climate science. While traditional approaches assume a linear vector autoregressive (VAR) model and are vulnerable to misspecification, we instead address a model-free version of the problem. We propose nonlinearly regressing both the outcome and exposure on the joint history of $Y$ and $Z$, and calculating a test statistic based on the sample covariance of residuals, we call the Generalised Temporal Covariance Measure (GTCM). To account for heteroscedasticity and improve power against local alternatives, we incorporate variance weights and employ a data-adaptive test based on polynomial lag expansions.The type I error control of the test relies on the relatively weak assumption that user-chosen regression procedures estimate conditional means at a sufficiently fast rate that is slow enough to accommodate nonparametric settings. By further assuming stability of the regression procedures and weak dependence in the time series, we can utilise the entire dataset to estimate the conditional means without splitting the time series into subsets.

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.

As a fundamental task across computational statistics, scientific computing and machine learning, sampling from high-dimensional probability distributions has received increasing attention in recent years. Numerous sampling algorithms have been proposed, among which underdamped Langevin Monte Carlo (ULMC) methods based on underdamped Langevin dynamics (ULD) have emerged as a class of efficient ones. In this work, we introduce a “universal" predictor-corrector formulation that bridges Euler-type, UBU-type and randomized schemes through different choices of method parameters. Notably, the “universal" integrator induces two novel classes of low-cost integrators, termed low-cost randomized integrators (LC-RIs) and low-cost UBU integrators (LC-UBUIs), as well as their exponential-free variants based on polynomial and rational approximations. The resulting new UBU-type and randomized schemes require only one gradient evaluation and two Gaussians per iteration, considerably reducing the number of gradient evaluations or Gaussians per iteration required by existing counterparts. Further, a general framework of long-time error analysis is developed for general discretization schemes in a probability metric. Under certain smoothness and non-log-concavity conditions, we rely on the unified framework to establish non-asymptotic $\mathcal{W}_1$-error bounds of both old and new schemes, revealing convergence rates of order $\mathcal{O}(d^{\frac{1}{2}}h)$ for Euler-type schemes, order $\mathcal{O}(d h^2)$ for UBU-type ones and order $\mathcal{O}(d^{\frac{1}{2}}h^{\frac{3}{2}})$ for randomized ones. In the strongly convex setting, the same non-asymptotic error bounds can be recovered in $\mathcal{W}_2$-distance. Numerical experiments corroborate the theoretical findings.

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.

We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.

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.

We study the Bayes-optimal spherical linear model as the ambient dimension and sample size grow proportionally, under a quantitative Marchenko–Pastur spectral-regularity condition on the design. This condition is satisfied by normalized i.i.d. designs with standardized entries of finite fourth moment, but does not require entrywise independence or impose conditions on the singular vectors. Under this condition, we prove a quantitative all-temperature TAP approximation and characterize the posterior geometry. For the natural finite-aspect-ratio TAP functional, the normalized spherical free energy and the TAP optimum differ by $O_P(p^{-1})$. Each is within $O_P(p^{-1/2})$ of its explicit deterministic equivalent, and this fluctuation scale is sharp. Uniformly over all global TAP maximizers, the normalized squared Euclidean distance to the spherical posterior mean is $O_P(p^{-1})$. We also prove that the posterior mass outside a data-dependent band determined by the ridge estimator has sharp exponential order. More precisely, uniformly over sufficiently small band widths $\varepsilon$, the logarithm of this mass is at most $-cp\varepsilon^2+O_P(1)$. For every fixed geometrically admissible width, a spherical-cap construction gives a matching exponential-order lower bound on this mass. For every deterministic sequence of widths $\varepsilon_p\gg p^{-1/2}$, the corresponding bands capture asymptotically all posterior mass.

In a statistical factor model, principal components (or eigenvectors) of a sample covariance matrix serve as estimates of principal directions, the true drivers of co-movement of a collection of observed variables. We write the often substantial error in these estimates as a sum of two interpretable terms, which we show have almost sure asymptotic limits as the number of variables grows with sample size bounded. This scenario is commonplace in financial economics, genomics, machine learning and signal processing. Out-of-subspace error measures the distance from an estimate to the subspace spanned by population factor exposures. It can be expressed in terms of data, providing an estimable floor for asymptotic error. In-subspace error is driven by a rotation error within that subspace, which arises from the fixed sample size of the latent factor returns and cannot be estimated from data alone. We illustrate our error analysis with a three-factor simulation of the US public equity market, showing the dependence of the magnitude of the error and its components on dimension and sample size. In that simulation, out-of-subspace error dominates. Researchers who rely on principal component analysis to estimate factor models can use our results to quantify errors in model-based predictions and attributions.

A popular selling point of diffusion large language models (dLLMs) is their capacity for parallelism: the ability to generate sequences of text far more efficiently than autoregressive models, which require one forward pass per token. Yet among the many competing paradigms for dLLMs, from masked to uniform to Gaussian diffusion, principled understanding of how these different proposals compare in parallelism remains limited. In this work, we initiate a fine-grained comparison of the capacity for parallelism among these three leading approaches and prove the following: - Uniform and Gaussian diffusion can sample in a number of forward passes which scales with the dual total correlation of the underlying distribution, a measure of intrinsic complexity which can be much smaller than the context length. Previously, it was only known how to achieve this using masked diffusion. - For a certain family of random empirical measures, we show that $\widetilde{\Theta}(\sqrt{d})$ forward passes are necessary and sufficient to sample using uniform or Gaussian diffusion, yet there exist approximate score oracles for which $\widetilde{\Omega}(d)$ forward passes are needed for masked diffusion. This establishes the first provable separation in parallelism between the three prevailing dLLM paradigms. Contrary to popular intuition that masked diffusions are harder to parallelize because they must commit to token values, the latter separation instead comes from the fact that the critical windows in masked diffusion sampling are asymptotically narrower than those in uniform and Gaussian diffusion sampling.