Papers

Clustering populations of networks while recovering their latent hierarchical organization is a fundamental yet largely unexplored problem in network analysis. To formalize this, we introduce the Hierarchical Distance Matrix, a specific class of population-level distance matrices that encodes latent hierarchical organization through recursively nested distance separation, accommodating unbalanced tree depths. Building on this framework, we propose a fully data-driven top-down procedure: network hierarchical clustering based on two-sample testing (NHC-TST). The algorithm recursively splits networks via spectral clustering and uses a graph-based two-sample stopping rule. The procedure adaptively determines the branching structure without requiring prior knowledge of the number of clusters or tree depth. Theoretically, we establish exact recovery of the population-level hierarchical structure and statistical consistency in the empirical procedure. Simulation studies demonstrate highly accurate recovery of both cluster memberships and hierarchical relationships across a wide range of settings. Applied to a global migration dataset, NHC-TST uncovers interpretable multi-resolution temporal structures that are not revealed by conventional flat clustering approaches.

We develop the Bayesian fusion forest, a nonparametric framework to estimate heterogeneous treatment effects on survival outcomes by combining a randomised controlled trial and real-world data. The framework relaxes the unconfoundedness assumption on the real-world data by assuming instead that the treatment effect transports across the two sources. Our method opens up right- and interval-censored outcomes to data fusion. We model the survival time with an accelerated failure time decomposition into a shared baseline prognosis, a source-specific deviation, a treatment effect, and a confounding function. The confounding function absorbs the confounding bias in the real-world data. Each component receives a Bayesian tree ensemble prior. The shared baseline prognosis borrows strength across sources, while the deviation captures between-source heterogeneity. A hierarchical Dirichlet process mixture models the error distribution nonparametrically. A simulation study shows efficiency gains over a trial-only analysis across varying levels of confounding and between-source heterogeneity. We combine the ACTG 175 trial with the Multicenter AIDS Cohort Study to estimate the effect of combination antiretroviral therapy for HIV. The fusion identifies a benefit for nearly every patient whereas the trial alone is inconclusive.

The Möbius transformation that generates the spherical Cauchy distribution from the uniform is the tangent-normal lift of a one-dimensional optimal transport: the monotone rearrangement of the cosine about the location axis. This identifies the probabilistic nature of the Möbius transformation and suggests a generalization: replacing the Cauchy target by any rotationally symmetric law, for instance the Poisson kernel or spherical cardioid, yields a generalized Möbius transformation. Möbius transport of a von Mises-Fisher base gives tractable anisotropic distributions on the sphere, with closed-form densities that inherit the base normalizing constant and allow immediate simulation. The Möbius-von Mises-Fisher and isotropic scaled von Mises-Fisher distributions, the latter also arising from a Möbius transport, are illustrated on paleomagnetic directions and short-period comet orbits, where they outperform classical and recently proposed alternatives.

We study expected improvement (EI) for minimizing a deterministic function $f$ in the RKHS $\mathcal H_k$ of a continuous positive-semidefinite kernel $k$ on a nonempty compact set $\mathcal X\subset\mathbb R^d$. Function values are observed exactly, and EI is computed from a fixed zero-mean Gaussian-process model with covariance $\sigma^2k$, $\sigma>0$. A weak-EI policy queries a point whose EI is at least a fixed positive fraction of its maximum. We introduce a notion of sequential separation radius relating ranked selected-point innovation norms to Kolmogorov widths, drawing on greedy approximation. Standard power-function estimates from scattered-data approximation and a finite-budget regret argument yield the rates. After $N$ post-initial queries, every weak-EI policy has simple regret $O(N^{-\nu/d})$ for isotropic Matérn kernels of smoothness $\nu>0$ and $O(\exp[-c_1\min\{N,N^{1/d}\log(eN)\}])$ for the isotropic squared-exponential kernel, with $c_1>0$. For $d=1$, the sharper bound $O(\exp[-c_2N\log(eN)])$ holds for exact EI, with $c_2>0$. These bounds are uniform over each fixed RKHS ball. If $\mathcal X$ has nonempty interior and $B>0$, the exact EI policy is minimax-rate optimal over the RKHS ball of radius $B$ for Matérn kernels, even among randomized strategies whose final recommendation need not be a query point. For the squared-exponential kernel, it is minimax-rate optimal up to constants in the exponent among deterministic methods whose final recommendation may be any point of $\mathcal X$.

We develop fast methods for conditional forecasting and structural scenario analysis with high-dimensional Bayesian vector autoregressions (VARs). Our general framework features a factor structure on the reduced-form errors, which enables fast and order-invariant equation-by-equation estimation; suitably identified factors admit a structural interpretation. The scenarios are defined through separate distributional restrictions on observables, structural shocks and idiosyncratic components. The computational cost of our proposed algorithm is cubic only in the number of restrictions, while the dimension of the forecasted system enters linearly. In our application with $33$ macroeconomic and financial variables and ten set-identified structural shocks for the US, we compute counterfactual predictions for oil price scenarios in the context of the 2026 closure of the Strait of Hormuz. The same oil price path is consistent with outcomes ranging from a mostly benign episode to pronounced stagflation, depending on which structural and idiosyncratic shocks are allowed to deliver it.

A strategy for incorporating higher-frequency determinants of inefficiency into the stochastic frontier models with production function indicators compiled at lower frequencies is proposed. Furthermore, the problem with positive skewness in stochastic frontier models is resolved by postulating a logistic function for the inefficiency equation and refraining from assuming a more complicated distribution of the inefficiency term. High frequency determinants are incorporated into the inefficiency equation in a nonparametric function. The model is estimated with hybrid methods in a backfitting algorithm instead of maximum likelihood estimation (MLE), which typically struggles with convergence issues and is vulnerable to distributional assumptions. The model is then used to characterize the linkage between firm efficiency and extreme weather events using data from electric cooperatives in the Philippines. This provides an approach to assess micro level sustainability issues from extreme weather arising events from climate change.

Classical clustering methods typically assume that all informative features support a single latent partition of the observations. This assumption can be overly restrictive for modern high-dimensional data, where different subsets of features may encode distinct notions of similarity and induce heterogeneous sample partitions, while some features may contain no meaningful clustering information. We develop a frequentist framework for local clustering that simultaneously identifies feature groups and estimates the sample clustering structure associated with each group. Our approach represents each sample partition by a label-invariant clustering matrix and groups features according to their shared clustering structures, thereby reformulating local clustering as a feature-grouping, or clustering-of-clusterings, problem. Under a heterogeneous sub-Gaussian mixture model, we construct feature-specific Gaussian-kernel similarity matrices and propose a local spectral clustering procedure based on a clustering-matrix optimization criterion. The proposed method avoids explicit likelihood specification and Bayesian posterior computation, accommodates heterogeneous feature distributions, and permits the presence of non-informative features. Extensive simulations and applications further demonstrate the practical utility and superiority of the proposed approach.

Bayesian analysis is increasingly used in clinical trials. However, assessment of the design with respect to the frequentist operating characteristics, such as type I error and power, remains a regulatory requirement in many cases. It is well established that, when information is borrowed from external sources to the trial, imposing strict frequentist type I error rate control is equivalent to offsetting the borrowing, which results in no power gains. We propose a Bayesian group sequential design with dynamic borrowing that exploits an explicit correspondence between Bayesian decision criteria based on posterior odds and frequentist uniformly most powerful (UMP) tests. At each interim analysis, two evidential thresholds are made available: the one that exactly retrieves the frequentist UMP decision; the other, that allows the investigator to incorporate historical information when appropriate. We assess the performance of the proposed approach in numerical studies, and apply the framework to the design of a phase III tuberculosis prevention trial, incorporating historical adult and pediatric trial data.

Metasignal is an open-source Python package for signal detection theory (SDT) and metacognitive measurement. It implements the 17 metacognitive measures evaluated by Rahnev (2025), together with the reference variables d' (perceptual sensitivity), response criterion c (response bias), and mean confidence. The 17 measures comprise three meta-d' family estimates, meta-d', M-ratio, and M-difference; four nonparametric Type-2 measures, the Type-2 area under the receiver-operating-characteristic curve (AUC2), Gamma, Phi, and delta confidence, together with their eight SDT-normalized ratio and difference forms; and two model-based measures, meta-noise and meta-uncertainty. A single function computes the complete set from trial-level stimulus, response, and confidence arrays. `metasignal` currently supports binary (two-alternative) discrimination tasks, in which each trial's stimulus and response are coded with exactly two categories. The package also provides a command-line interface, group summaries, bootstrap confidence intervals, permutation tests, optional hierarchical Bayesian models, and information-theoretic measures. `metasignal` unifies these measures in a single platform to encourage broader metacognition research and adoption in decision-making studies.

Counterfactual explanations (CEs) enhance the interpretability of machine learning models by identifying the smallest change to an input required to obtain a desired output. Although CEs are conventionally formulated as a distance-minimization problem, the theoretical basis of this formulation has received limited attention. We show that a distance-minimization-based CE is mathematically equivalent to the maximum a posteriori (MAP) estimate of a Gibbs posterior within the generalized Bayes framework, specifically when a distance-based prior is used. We call this formulation the Distance-Prior Generalized Bayes CE (DP-GBCE). Building on this posterior perspective, we introduce two decision rules beyond MAP within a unified framework: a Bayes decision that minimizes expected decision loss and CVaR-CE, a risk-averse decision rule. We also propose an extension that uses Bayesian model weights to mix the posterior distributions of multiple models, thereby accounting for model multiplicity, where several models have comparable predictive performance. Finally, we define metrics for evaluating both individual CEs and the posterior distribution as a whole, and use experiments on simulated data and Google Trends data to quantify the trade-offs among the decision rules.

Modern opaque AI models prize performance over interpretability, which makes testing difficult. However, formal statistical tests conducted on a model's embedding space can provide robust characterizations of semantic structure, concept separation, and knowledge graph alignment. Model developers would benefit from a model comparison technique that leverages human-curated knowledge structures to test alignment. The scale of the input space for even relatively simple tasks motivates the need for alignment checks that augment standard outcome reasoning. This work develops and demonstrates a topology-based multi-modal alignment test to make deployment, selection, and comparison of opaque models more interpretable. These methods also offer an intuitive connection to possibility theory and a unified decision theoretic framework from data to deployment.

We realize the GML estimator by an expectation-maximization algorithm initialized near the true positions and verify the theory by simulation.

Testing independence or conditional independence is fundamental to statistical inference, yet existing methods for non-Euclidean random objects often face a difficult trade-off between geometric flexibility and theoretical tractability. We introduce the Distance Profile Embedding (DPE), a novel representation that maps random objects from general metric spaces into a Hilbert space of square-integrable functions. We prove that this mapping is injective and preserves full distributional information without requiring isometric Hilbert embeddings or one-to-one correspondence conditions. Leveraging the DPE, we develop a unified framework for marginal and conditional independence testing of random objects that enjoys a rigorous asymptotic theory for both size and power. Notably, our framework is the first in the literature to accommodate object-valued conditioning variables when testing conditional independence, overcoming the Euclidean or Hilbertian constraints of existing methodologies. We facilitate the calculation of analytic $p$-values using closed-form asymptotic null distributions, which avoids the computational burden of permutation tests common in existing metric-based methods. The numerical properties of our methods are demonstrated through both simulations and two real-world applications involving gut microbiome compositions and global human mortality distributions, respectively.

I present a very simple example in which a full-support prior over distribution functions has an inconsistent posterior, which I believe has instructive value. The example is $Y_i \stackrel{\text{iid}}{\sim} P_0$ under the Bayesian hierarchical model $[Y_i \mid P] \stackrel{\text{iid}}{\sim} P$ and $P \sim \int \mbox{DP}(\alpha, H) \, \pi_\alpha(\alpha) \ d\alpha$ when $\pi_\alpha(\cdot)$ has exponential (or heavier) tails, where $\mbox{DP}(\alpha, H)$ denotes a Dirichlet process with concentration parameter $\alpha$ and mean $H(\cdot)$; on the other hand, consistency is obtained under a light-tailed prior. This unifies and generalizes a consistency result of Freedman and Diaconis (1983) with an inconsistency result described by Ferguson et al. (1992).

For comparing two survival curves at a fixed timepoint with right censoring, a standard method uses asymptotic methods on two Kaplan-Meier estimators with Greenwood variance estimators and transformations using the delta method. Confidence intervals on associated estimands can either breakdown or undercover due to Kaplan-Meier estimates of zero or one, small sample sizes, or heavy censoring. Although others have proposed adjustments to the Greenwood variance for a single sample, these adjustments have not been incorporated into the two-sample application of the delta method and we do that here as well as provide modifications when Kaplan-Meier estimates are zero or one. An alternative non-asymptotic existing method called melding on the beta product confidence procedure (BPCP) appears to have at least nominal coverage regardless of the sample size. Without censoring the problem reduces to comparing two independent binomials and melding on the BPCP gives p-values equivalent to Fisher's exact test p-values and further gives compatible confidence intervals for treatment comparison estimands; however, the melding can be very conservative. In this paper, we expand the melding on the BPCP intervals to include mid-p versions, which are designed to achieve coverage closer to nominal without guaranteeing coverage for all cases. We study these existing methods and our minor modifications of them primarily by numerical methods or simulation. Results suggest only the melding on the BPCP can guarantee coverage in all studied scenarios, and the mid-p version most often has closest to nominal coverage. We provide R functions in the bpcp R package.

Surveys such as DESI release data in stages, and each release invites a fresh assessment of whether dark energy is consistent with a cosmological constant. Standard Wilks-based $\sigma$ values are interpreted as if the test were performed only once, yet revisiting the same question at DR1, DR2, DR3, and beyond inflates the chance of an apparently significant fluctuation. We reanalyse the DESI BAO evidence for dynamical dark energy using an e-process, a likelihood-ratio-based measure of evidence that remains valid under repeated looks at accumulating data: it controls the false-detection probability under continued testing across releases with no trials-factor penalty. Applied to the DESI DR1$\to$DR2 BAO sequence, the result depends strongly on which departures from $\Lambda$CDM the test is designed to detect. For a pre-specified alternative aligned with the DR2-preferred direction, the running evidence reaches $M_{DR2} = 33.97$, crossing an illustrative 5% threshold of 20. The evidence is concentrated almost entirely in the LRG2 redshift bin, and removing that bin reduces the running e-value to $M = 0.49$, mildly favouring $\Lambda$CDM. When allowing any of the seven bins to have produced the excess, the signal does not survive a look-elsewhere correction. Other physically agnostic alternatives also fail to reject $\Lambda$CDM, while a physically motivated thawing-quintessence alternative rejects more strongly than any agnostic choice. The concentration of evidence in LRG2 specifically is difficult to reconcile with a smoothly evolving equation of state. The data therefore points to a fragile, single-bin, specification-dependent signal rather than robust evidence for dynamical dark energy. We recommend that future DESI, Euclid, Roman, and LSST data releases report an anytime-valid e-process value with its test specification stated alongside conventional $\sigma$ significances.

Variance-based global sensitivity analysis (GSA) plays a key role in uncertainty quantification by identifying the contributions of uncertain inputs to the variability of the model response. The repeated model evaluations required for these tasks are often prohibitively expensive; surrogate models provide an efficient alternative by constructing inexpensive approximations of the underlying system response. Constructing surrogate models that combine scalability and interpretability for systems with high-dimensional stochastic inputs and functional responses remains challenging, particularly when sensitivity estimates are required across spatial or temporal domains. Polynomial chaos expansion (PCE) provides an effective framework for uncertainty propagation and sensitivity analysis due to its orthogonal structure and direct relationship with variance-based sensitivity measures. However, PCE suffers from the curse of dimensionality, whose computational burden is amplified for problems with functional responses. In this work, we introduce the Structured Neural Chaos (sNC) expansion as a surrogate modeling framework for variance-based GSA, inspired by the interpretability and orthogonal structure of PCE. The proposed framework retains the interpretability of structured decompositions while leveraging the expressive power of neural networks. The sNC expansion mirrors a truncated functional ANOVA decomposition, where each interaction component admits a separable low-rank approximation whose basis functions and coefficients are parameterized by neural networks. The expansion is constructed sequentially, adaptively identifying the dominant modes within each ANOVA subspace and determining the effective complexity of the representation. The resulting structure enables the extraction of statistical and sensitivity quantities directly from the coefficients of the sNC expansion at negligible cost.

We show that, for a given dimension, the set of Spearman's rank correlation matrices and that of linear correlation matrices coincide if and only if the dimension is no larger than nine. For this, we construct an extreme rank-four counterexample in dimension ten and prove its incompatibility using moment identities and Cauchy-Schwarz. Appending unit directions produces counterexamples in every higher dimension. This, together with existing results, completes the dimensional classification and settles a long-standing open question in quantitative risk management.

We investigate penalized likelihood methods for estimation and inference in conditional logistic regression. The standard conditional maximum likelihood estimator is known to be biased away from zero in small or sparse matched case-control studies. A widely used remedy is Firth's penalized likelihood approach, which has good frequentist operating characteristics but provides limited control over the degree of shrinkage applied to individual regression coefficients. We develop point and interval estimators by penalizing the conditional likelihood with independent log-$F$ distributions. The log-\(F\)-penalized approach allows analysts to calibrate shrinkage using interpretable prior assumptions about plausible effect sizes. We also provide practical guidance for calibrating the amount of shrinkage and show that the method can be implemented through data augmentation using standard conditional logistic regression software. We illustrate the methods using data from (i) a study of maternal exposure to diethylstilbestrol and the risk of vaginal cancer in daughters, and (ii) a genetic association study of type 2 diabetes. We then compare the log-$F$-penalized approach with Firth's penalized likelihood method in a simulation study. In simulations, the log-$F$-penalized estimators had confidence-interval coverage comparable to that of Firth's method and lower mean squared error, with similar type~1 error rates and power. These results support the use of log-$F$-penalized conditional logistic regression for inference in sparse matched and stratified studies.

Computational cognitive modeling seeks to infer latent cognitive mechanisms underlying observed behavior. Bayesian inverse planning provides a principled framework for such inference, but its success depends critically on the experimental environment. Existing approaches typically treat environments as fixed, leaving open the question of which cognitive experiments are most informative for cognition parameter inference. We formulate the design of cognitive planning experiments as a Bayesian Experimental Design (BED) problem, treating the experimental environment as the design variable. We establish an exact Monte Carlo BED benchmark and introduce an amortized Bayesian experimental design framework for efficient posterior inference and design evaluation. Experiments on the Mouselab-MDP process-tracing paradigm show that amortized BED closely matches the environment rankings of exact Monte Carlo BED while substantially reducing computational cost. We further show that no single environment is uniformly optimal across cognitive inference objectives, revealing trade-offs between expected information gain, posterior recoverability, and information efficiency. These results provide a principled framework for designing informative cognitive experiments for Bayesian parameter inference.

Simulation-based calibration aims to infer unknown parameters of complex simulation models by aligning model outputs with real-world observations. When simulation runs are computationally expensive, statistical emulators trained on simulation data are used to efficiently approximate the model. An intelligent, adaptive selection of simulation inputs for building the emulator can substantially improve the efficiency of the calibration process. This task is particularly challenging for stochastic simulations with noisy outputs, since both selecting new input locations (exploration) and allocating repeated runs at existing inputs (replication) are essential for efficiently learning the input-output relationship. In this paper, we introduce an active learning framework that adaptively balances exploration and replication for data-efficient calibration. Our uncertainty-aware acquisition criterion targets learning the posterior density of the unknown simulation parameters, and we derive two corresponding forms of the acquisition function for exploration and replication. Building on these, we propose a strategy that, at each stage of the sequential design, chooses between exploration and replication to most effectively reduce the uncertainty in the estimate of the posterior density of the simulation parameters. Experiments on synthetic benchmarks and a real epidemiological model demonstrate that our approach significantly improves learning of the posterior distribution of the simulation parameters while reducing the number of required simulations, making it well-suited for expensive stochastic simulation settings.

Tree-based diffusion models fit flexible conditional predictive distributions for tabular regression without a neural density estimator, but they inherit their design defaults—noising path, parameterization, training distribution, features, sampler—from the neural setting. We show these defaults are the binding constraint: what a gradient-boosted ensemble actually solves is a supervised regression problem whose conditioning they determine. We present DiffGBM, which makes them explicit along two axes. First, a Gaussian-path flow-matching trainer for $p(y \mid x)$ that learns a velocity field directly and recovers the score algebraically, admitting few-step deterministic ODE sampling. Second, we expose the score-side recipe—residualization, EDM-style preconditioning, log-sigma time sampling, noise-level features, loss weighting, and histogram resolution—as jointly tunable axes over a shared LightGBM surface rather than one frozen bundle. This score-flex space represents the published recipe as a special case; across eleven tabular benchmarks under fold-0 tuning, folds-1–5 evaluation, and a matched 40-trial budget and sampler, the selected configurations beat that baseline on every dataset (paired Wilcoxon $11/0$, $p<10^{-3}$), with the best aggregate CRPS skill (0.725 vs.\ 0.699) of any row. The two rows are complementary: score-flex buys accuracy with a stochastic sampler and is the slowest row, while flow matching is the cheapest sampler ($5.2\times$ faster than the published baseline) and the best-calibrated DiffGBM row. Tuned non-diffusion baselines still win individual datasets, and stochastic ($\varepsilon>0$) flow samplers do not Pareto-dominate the deterministic corner.

The parametric score test assesses a hypothesis through derivatives of the log-likelihood, whose expectation vanishes under the null. When the parameter of interest is a regression function identified as a risk minimiser, we extend this idea to test whether it belongs to a given linear function class. This yields goodness-of-fit tests for common semiparametric regression models, including generalised additive and partially linear models. Suitably formulated, the framework also detects effect modifiers in observational studies. We propose a hunt-and-test strategy that splits the data into two: on one part, after fitting the null model, machine learning is used to identify a promising direction in the empirical scores; on the other, we test whether the score vanishes in that direction. To account for error in estimating the null model, we apply a debiasing correction based on a weighted least squares projection. We establish Type I error control under relatively mild conditions and show the test has power whenever the hunted direction is correlated with the true score. Simulations and real-data examples demonstrate favourable performance, including identifying effect modifiers in an HIV clinical trial and assessing an additive model for insurance claims. The methodology is implemented in the R package dScoreTest.

Algorithmic credit scoring must satisfy fairness and explanation requirements, yet prevailing predictive-parity criteria assess only outcomes at the decision point. They can therefore overlook whether rejected applicants face unequal burdens in reaching future approval, a phenomenon we call masked inequality. We develop an effort-centric framework that measures an applicant's effort as the minimum weighted cost of feasible changes required to cross the approval boundary. The framework distinguishes feature-independent actions from additive structural shifts that propagate through a causal model and defines parity by comparing average minimum effort across protected groups. We derive tractable local expressions for general differentiable classifiers and exact expressions for logistic regression, embed them in an in-processing fairness objective, and bound changes in portfolio credit risk. The same optimisation yields actionable pathways to approval. Using mortgage data with continuous and discrete features, we find that rejected female applicants require greater effort even when standard predictive-parity criteria are satisfied. Feature-independent regularisation reduces the effort gap by more than 50% with modest predictive changes. Causal regularisation yields reductions above 90% at the tested positive penalty weights, but with larger predictive and risk-return trade-offs. Expected and unexpected losses remain broadly stable under feature-independent regularisation and increase under causal regularisation; RAROC declines but remains positive. These results show that effort parity complements predictive fairness by revealing and mitigating hidden barriers to future credit access while making the associated operational trade-offs explicit.

Assessing a single model fit requires a computable upper confidence bound for the gap between the fit and the unknown truth, as mean estimates ignore realization variance. Standard cross-validation margins are bottlenecked at order $n^{-1/2}$ by noise fluctuations, even when the true error shrinks faster. While wild refitting cancels this noise level, existing Rademacher sign methods degenerate for kernel ridge regression and rely on unobservable quantities. We propose a Gaussian refit for kernel ridge regression. By Anderson's inequality, the fit movement is monotone in the noise sizes, yielding a computable tail bound. Assuming only symmetric noise, the bound requires no moment assumptions and is calibrated at any confidence level via order statistics. Theoretically, using a worst-case envelope, the bound contracts at the minimax rate $O_P(n^{-2s/(2s+1)})$, correctly matching the prediction error. Empirically, using a practical data-driven envelope, the bound maintains full coverage within twice the true $95\%$ error quantile. By contrast, cross-validation exceeds this quantile by factors up to $51$, and by hundreds under infinite-variance noise. The procedure extends empirically to nonlinear constrained estimators and real spatial data.

Bayesian Additive Regression Trees (BART) have shown state-of-the-art performance in both prediction and causal inference problems. Previous theoretical work has attempted to explain BART's superior performance by establishing posterior contraction rates for standard BART models, but these rates depend strongly on the number of covariates. Here, we take a different approach and study the behavior of BART as the number of trees grows towards infinity. We show that in this regime, BART converges to a Gaussian process (GP) with a particular kernel. The kernel and its corresponding reproducing kernel Hilbert space (RKHS) have favorable inferential properties that help explain BART's excellent performance. We introduce random tree features as an approximation to this limiting GP, and establish minimax-optimal learning rates for ridge regression on these random features that depend only logarithmically on dimension. In addition to providing insight into the empirical success of BART, random tree features offer a computational benefit over traditional MCMC estimation. The random-features approximation also allows practitioners to easily incorporate BART into any model which has a linear predictor, expanding the applicability and flexibility of BART.

The value of an individualized treatment rule (ITR), defined as the expected outcome under treatment assignment according to the rule, is useful for assessing average clinical benefit but does not explain how the benefit of a rule is generated. We propose a causal mediation framework for decomposing the value contrast between a prespecified candidate ITR and a clinically meaningful reference rule into direct and indirect components. Using rule-specific nested potential outcomes, we define natural direct and indirect rule effects that quantify the extent to which the improvement in value arises through pathways operating directly on the outcome versus through a specified mediator. We give identification conditions under which these components are identified by a rule-level mediation g-formula. For estimation, we adapt Bayesian causal mediation forests to obtain posterior inference for the value contrast and its path-specific components. Our simulations demonstrate that the proposed estimator achieved near-nominal credible interval coverage with decreasing bias and root mean squared error as sample size increased in settings with varying direct and mediated contributions. We further illustrate the method using data from the TRIUMPH trial, decomposing the cognitive benefit of a lifestyle intervention rule through candidate neurovascular, cardiorespiratory, and behavioral mediators. The proposed framework complements optimal ITR learning with explanation using mediation, providing a natural approach for mechanistic evaluation of ITRs.

The test-negative design (TND) is widely used to estimate vaccine effectiveness (VE) for respiratory pathogens by comparing vaccination odds among test-positive cases versus test-negative controls. A central yet underexplored design element is which test-negative illnesses constitute valid controls. With rapid multiplex PCR panels, investigators can now identify specific non-focal pathogens among test-negative patients, allowing for better characterization of “test-negative illness”, but also revealing a mixture of control outcomes that may each satisfy or violate causal assumptions. We synthesize recent causal identification results for the TND and show that they imply two distinct interpretations of control selection: 1) a sampling view in which controls represent the source population and 2) a bias-correction view in which controls function as negative control outcomes under equi-confounding. Building on these interpretations, we develop a framework for multiplex-informed control selection. We propose a taxonomy that distinguishes controls that serve primarily as exposure proxies (sharing unmeasured determinants of infection) from those that serve as testing proxies (sharing unmeasured determinants of care-seeking), derive implications for pathogen-specific and pooled estimators, and suggest three practical principles for control selection: vaccine irrelevance, avoidance of entanglement with other interventions, and testing-process comparability. We also formalize nuances introduced by multiplex panels, including co-detections and pan-negative episodes, and outline when standard pooled estimators remain valid versus when alternative estimators are needed. In simulations across 9 scenarios, we demonstrate violations concentrated in a single control pathogen can substantially bias pooled TND estimates, whereas a pre-specified pathogen screening estimator remained unbiased.

Held-out log-likelihood is the standard currency for comparing statistical models of neural spike trains, and is often reported as bits per spike relative to a homogeneous Poisson baseline. The units of this metric are difficult to reason about: it is rarely obvious whether an improvement of, say, $0.34$ bits per spike is a large effect or a negligible one. This note develops an interpretation of held-out log-likelihood borrowed from game-theoretic statistics. If $L$ denotes a model's expected log-likelihood ratio to the homogeneous Poisson baseline, then we show that $-\log(\alpha) / L$ represents the number of heldout time bins of recording needed to reject the baseline at level $\alpha$ under a particular null hypothesis testing procedure based on a betting game. Thus, a simple re-scaling of the log-likelihood yields an intuitive metric of model performance in units of recording time which answers "how much held out data would I need, on average, to disprove the null." We illustrate the construction on head-direction cells recorded in mouse anterior thalamus, where a generalized linear model reaches significance in roughly $120$ ms of held-out data for a strongly tuned cell and roughly $11$ s for a moderately tuned cell.

As image generation models scale to ever higher resolutions, global coherence, local detail, and texture fidelity become critical axes for generation quality. However, standard flow matching treats all spatial frequencies uniformly, ignoring the natural frequency hierarchy where high-frequency bands become indistinguishable from pure noise far earlier than coarse structures. We introduce WaiT, a Wavelet-aware image Transformer that decomposes generation into coarse and fine bands via lossless wavelets. True to its name, the high-frequency bands wait for the signal: staying pure noise until coarse structure has emerged, then joining the flow for joint refinement. Since standard FID discards fine-grained detail through aggressive downsampling, we introduce a more stringent three-axis evaluation protocol to assess quality at native resolution. On ImageNet 512x512, WaiT achieves a pixel-space FID of 1.43 and is Pareto-optimal across all three axes, reducing sampling compute by up to 50%. With our largest 2B model, we set a new state-of-the-art FID of 1.3 for pixel-space models on ImageNet 512 resolution. Our formulation outperforms even the strongest latent-space models on texture fidelity, and scales seamlessly to high-resolution OpenImages and to video generation, achieving a state-of-the-art FVD of 0.84 on Kinetics-600 with no algorithmic modifications.