Papers

System One models output calibrated probabilities over typed answers such as categorical choices, ordinal levels, or binary outcomes, via a non-generative interface. Software can act on these probabilities through thresholds, cost-weighted choices, and escalation rules. Consequently, if these probabilities are miscalibrated or wording-sensitive, the software ma take unintended actions leaving human operators with no textual rationale to inspect. Current evaluations largely report accuracy and calibration on public classification datasets without a clear reference. We present TypedBench, a benchmark for typed decision models like Jev built from seven policy-labelled generators and nine evaluation suites. We report accuracy as median and range across paraphrases, and calibration error relative to the finite-sample noise floor of a matched, perfectly calibrated predictor. We assess probability quality through selective prediction, ordinal proper scoring rules, and realised cost under asymmetric cost matrices. We evaluate a hosted model, an open encoder, and a family of open decoders spanning 0.8B-9B parameters on identical items. The hosted model follows the stated policy but is wording-sensitive and systematically underconfident; under asymmetric costs, using its probabilities can be worse than taking its top answer. Decoders route exactly and are slow as options or questions are added. The decoder is least accurate on policy questions and degrades with more options. Overall, typed decision models must be evaluated jointly on policy adherence, wording robustness, probability quality, and induced decision outcomes.

Conditional independence testing is a ubiquitous problem in scientific discovery. The widely employed model-X assumption shifts the modelling burden from the dependence of the output on the inputs to the dependencies within the inputs. Log-optimal e-variables have been studied in this setting, but it remains unclear how to incorporate machine learning models into their design. Other approaches test exchangeability directly, yielding an e-variable with lower power in theory but, surprisingly, higher power in practice. We explain this phenomenon by decomposing the error into null enlargement, approximation, and estimation error. The decomposition shows that GRO e-variable estimates can be beaten because of their worse approximation and estimation errors, and we explore intermediate null hypotheses between model-X conditional independence and exchangeability to reduce these errors. Moreover, the model-X assumption often only holds up to an estimation error, invalidating exact type-I error guarantees. We provide estimation error bounds that accommodate triple robustness results, achieving fast convergence rates.

Generative conformal prediction builds uncertainty sets from samples of a conditional generator, which are efficient only when the samples represent the response distribution well. This can require many samples, each of which can be costly, as in large diffusion models and scientific simulators, so the sampling budget must be used efficiently. Existing methods draw the same number of samples at every input, wasting samples where the response distribution is simple and undersampling where it is complex, which inflates sets and leaves those inputs under-covered. We propose CASA (Conformal Adaptive Sample Allocation), which characterizes the marginal value of an additional sample and allocates samples across inputs to minimize the expected set size subject to marginal coverage and an average sampling budget. Theoretical analysis shows that adaptive allocation yields smaller sets than a fixed count at the same budget: a missed mode forces a radius that spans the gap between modes, and even oracle radius cannot compensate for it. On synthetic and real tasks, CASA produces substantially smaller sets at the same budget, often improves conditional coverage, and complements existing radius-adaptive methods.

Estimating covariate-dependent intensity functions from spatial point pattern is challenging when the integrated intensity function cannot be evaluated analytically. This challenge becomes even worse in Bayesian inference due to repeated integration of the intensity function over the high-dimensional covariate space. This paper develops a Bayesian soft-tree Poisson process model that combines adaptive covariate partitioning with smooth intensity function. The key idea is to use soft gating functions to smooth transitions between partition boundaries in a tree generating process while preserving analytic evaluation of integrated intensity. A Bayesian CART prior is adopted to learn the partition structure from data. Given the partition tree, conjugate priors are assigned on leaf parameters to enable analytic marginal likelihood computation. This paper also develops a reversible-jump Markov chain Monte Carlo (RJ-MCMC) algorithm for posterior inference. Posterior consistency is established for the proposed model with respect to the Hellinger distance. Extensive simulation studies and an application to wildfire data confirm the advantages of the proposed method over existing methods.

We provide the first generic exact sampling algorithms with finite expected total cost for uniformly elliptic multivariate diffusions and the corresponding diffusion bridges with bounded coefficients and bounded continuous first derivatives. We formulate a computational cost model that counts work in terms of generation of Gaussian and uniform random variables to execute acceptance–rejection sampling, as well as function evaluations of the drift and diffusion coefficients, together with elementary operations such as sums and multiplications. Our constructions combine randomized parametrix expansions with a finite weighted graph that incorporates both bridge endpoints into the proposal. The exact sampling of the diffusion is minimax optimal jointly in the horizon and number of requested observations, while bridge sampling is minimax optimal in the number of observations for fixed horizon and endpoints.

Different optimizers can fit the same training data while selecting classifiers with substantially different geometries, but whether this difference provably affects population performance remains unclear. We show that row-wise normalization can achieve strictly higher population accuracy than full-batch Adam, a proxy for random-reshuffling Adam, and exact-SVD Muon in high-dimensional multiclass classification. Under an isotropic Gaussian-cloud data model, this advantage arises because row normalization's class-wise Euclidean geometry asymptotically preserves the population decision-boundary directions, whereas Adam's coordinate-wise geometry and Muon's spectral geometry introduce nonvanishing distortions. Beyond isotropy, the advantage persists for full-batch training on class means with independently oriented class-mean and test-noise covariances. It holds for power-law spectra with class-mean exponent below one, even under heavily anisotropic test noise. When both covariances are diagonal and sufficiently close, the advantage over Adam can reverse, while applying the same random rotation to both restores it by changing only their alignment with Adam's coordinate axes. Synthetic and last-layer language-model experiments support the predicted advantage.

Autonomous scientific discovery with LLMs requires generating and testing hypotheses adaptively as evidence accumulates while maintaining statistical validity. Existing anytime-valid methods can handle data-dependent hypotheses, but open-ended discovery poses a deeper challenge: the best discovered hypothesis may still be the best of a bad lot, with better explanations yet undiscovered, while even background knowledge such as physical laws may require revision in light of new findings. In response, we formalize the problem as Abductive Autonomous Scientific Discovery (AASD) using possibility theory. We introduce abductive utility, a computable measure of discovery progress, and possibility frontier search, the first algorithm for AASD, which maintains anytime validity and achieves $\varepsilon$-optimal abductive utility asymptotically under suitable conditions. Experiments on synthetic and real-world scientific-discovery tasks show strong performance.

Large language models (LLMs) are commonly post-trained against LLM judges and other cheap surrogates because the true reward, such as human preference, is too expensive to query at scale. This practice often leads to reward hacking, where reinforcement learning against a miscalibrated surrogate leads to undesirable side effects. In this work, we study a setting in which a small number $n$ of model outputs are annotated with ground-truth labels (e.g., from expert review) and used to recalibrate the LLM judge before optimizing against it. Prior approaches to judge recalibration are costly or heuristic, and it is known that on-policy sampling fails when the surrogate is miscalibrated on a rare set of outputs. In this work, we propose envelope sampling, a theoretically-grounded method for judge recalibration that seeks to minimize an upper bound on the regret of the post-trained model under the assumption that the human reward and re-calibrated reward lie in an $L^2$ ball around the judge. We give practical algorithms to sample from the envelope by rejection or by fine-tuning against a modified reward, and experiments on clinical note generation and on a controlled sycophancy task show that recalibrating on envelope samples mitigates reward hacking where recalibrating on base-model samples does not.

The recent e-closure method can recover every procedure that controls FDR (and other expectation losses). But the recovered e-collection is “self-referential” and gives no insight on how to improve the procedure (if improvable), and some recent improvements have been somewhat opaque. We introduce an elementary new technique called anchoring that exploits looseness in existing FDR proofs to enlarge a baseline multiple-testing procedure's self-referential local e-value. The resulting e-closure thus transparently retains every baseline discovery (and usually adding more) and controlling the false discovery rate under the same conditions as the baseline. To show that this principle is broadly applicable, we use it to improve a large suite of multiple testing procedures: (i) Anchored-BH dominates the Benjamini-Hochberg (BH) procedure under PRDS while being incomparable to Goeman's recent closed-BH, (ii) Anchored-BY dominates the Benjamini-Yekutieli (BY) procedure under arbitrary dependence while being incomparable to closed-BY, (iii) For two-sided Gaussian p-values (under appropriate covariance conditions), Anchored-2BH dominates running BH twice at half the level on two one-sided p-values, (iv) Anchored-dBH dominates dependence-adjusted BH, (v) Anchored e-BH dominates e-BH and is incomparable to closed e-BH, (vi) Anchored SeqStep+ improves the original (including selective and adpative variants) while preserving ordered rejection structures. All of these are accomplished in sorting or quadratic time. The appendix shows how to dominate Shifted-BH (for two-sided arbitrarily correlated Gaussians) and NDBH (for negatively dependent p-values), IndBH (for graph-dependent p-values), some online FDR procedures and the support line procedure for boundary FDR control.

Small area estimation (SAE) often requires both borrowing information across areas and benchmarking estimates to reliable aggregates. We develop a Bayesian framework that addresses these two objectives jointly through a new family of Benchmarking priors. The priors are induced by a benchmark-constrained regularization problem. The resulting family includes a Benchmarking Prior that incorporates the benchmarking restrictions without additional regularization across areas, and Single and Multi-View Laplacian Benchmarking Priors that introduce regularization through graph Laplacians constructed from area similarities based on external covariate information. For posterior computation under these degenerate priors, we develop tailored MCMC algorithms based on a reduced parameterization of the constraint space. We assess the proposed models using a data-based simulation and apply the framework to estimate Average Household Size (AHS) at the municipality level in Colombia in 2025. In this application, graph-based regularization improves model performance, with the Multi-View models generally producing more precise municipality-level estimates.

Continuous structural connectivity describes the connection between any two points of the cortical surface by an intensity function. Based on the endpoints of tractography streamlines, existing methods estimate this function mostly by kernel smoothing, which only gives a point estimate. We propose a Bayesian framework that estimates continuous connectivity with uncertainty quantification. The endpoints of tractography streamlines, mapped to two spheres, are modeled by a log-Gaussian Cox process on the product of the spheres with a Matérn Gaussian process prior. With the finite element representation of the prior, the Laplace approximation of the posterior enables the computation on a fine mesh with over 400,000 latent variables per hemisphere pair. Numerical experiments show the accuracy of the point estimate of this approach and the coverage of its credible intervals. In the Adolescent Brain Cognitive Development Study, the proposed method predicts held-out streamlines better than kernel smoothing.

Testing equality of two high-dimensional covariance matrices is challenging when many entries differ only slightly. Dependence among sample covariance entries can also affect the finite-sample size and power of tests that aggregate their differences. We propose a studentized $\ell_2$-type statistic that averages squared, marginally standardized differences between corresponding entries of the two sample covariance matrices. Before constructing the statistic, we apply the same nonsingular linear transformation to both samples. In the transformed coordinates, the covariance-equality hypothesis is unchanged, whereas the standardized differences and correlations among their estimators generally change. The transformation can therefore incorporate structural or scientific information without reducing dimension. We approximate the null distribution using a Gaussian multiplier bootstrap that uses coordinatewise studentization and avoids forming or inverting the full covariance matrix of the vectorized sample covariance entries. For deterministic transformations, we derive nonasymptotic Gaussian and bootstrap approximation bounds and establish asymptotic size validity and power consistency. We also show that replacing a population transformation by a same-sample estimator leaves the statistic and bootstrap critical value asymptotically unchanged in relative terms under an operator-norm convergence condition. Simulations show that suitable transformations improve size accuracy and power under weak dependence. An analysis of breast cancer gene-expression data illustrates the method.

Weight-only post-training quantization (PTQ) relies heavily on reconstruction loss minimization to preserve model quality at low precision. We show that the weights favored by minimizing this loss need not yield better model performance on new tasks. In fact, we find that lower reconstruction loss can even degrade model performance on the same calibration data. Our analysis further shows that weights with lower reconstruction loss on calibration data can have higher loss than other weights when the distribution of input activations changes. Motivated by these observations and our analysis, we propose Distributionally Robust Quantization (DRQ), a post-hoc refinement process that minimizes worst-case reconstruction loss over a constrained set of input activation distributions. DRQ refines the integer codes representing quantized weights within the existing quantization grid, keeping quantization parameters and inference operators unchanged. Extensive experiments show that DRQ improves models quantized by six representative PTQ methods, including AWQ, GPTQ, and ParoQuant, and delivers gains across both dense and mixture-of-experts large language models. These results establish DRQ as a general post-hoc refinement framework for weight-only PTQ, achieving better downstream performance without adding inference overhead.

Hourly precipitation forecasting involves several distinct statistical challenges, including spatially varying predictor-precipitation relationships, a strongly imbalanced precipitation distribution, and progressive degradation of precipitation event skill with increasing lead time. We develop a spatiotemporal forecasting framework that addresses these challenges through three complementary components: explicit geographic representation, event-aware learning for imbalanced precipitation, and weak asymmetric regularization derived from the atmospheric water budget. The physical information is treated as an asymmetric constraint rather than as an additional prediction target, designed to discourage physically unsupported precipitation attenuation without replacing the data-driven forecast. Experiments using ERA5 data across regional, enlarged domain, and spatial subset settings show that explicit geographic information improves spatial field prediction, while event-aware learning provides the most consistent gains in detecting moderate and heavy precipitation events. Physical regularization has a more selective effect, mainly reducing systematic underprediction over the enlarged domain while improving longer lead precipitation event prediction in the spatial subset experiment. These results indicate that the benefit of physical guidance depends on the available data regime and becomes most apparent when data-driven precipitation information deteriorates with increasing lead time.

Functional generalized linear models provide a flexible framework for relating a scalar response to functional and scalar predictors, but their conventional formulation requires specification of the response distribution and is typically developed for a single population. We propose a semiparametric functional generalized linear model for multi-population data that leaves the baseline response distributions unspecified and links them across populations through a density ratio model. The proposed framework accommodates both functional and scalar predictors while borrowing information across related populations. We develop a maximum empirical likelihood estimation procedure and use penalized B-spline approximations to estimate the functional coefficients. A cross-validation procedure based on a kernel-smoothed empirical likelihood density estimator is introduced to select the smoothing parameters. Simulation studies show that borrowing information through the density ratio structure can improve estimation efficiency relative to fitting each population separately, and that the proposed semiparametric method remains competitive when a parametric Gaussian model is correctly specified while providing substantial robustness when it is misspecified. We illustrate the method using county-level soybean yield data from Kansas, with daily temperature curves and irrigation levels as predictors.

E-values provide a flexible framework for statistical inference, but the universal threshold $1/\alpha$ can be conservative when the null distribution has additional structure. We study how structural restrictions limit the worst-case concentration underlying Markov's inequality, using a nondecreasing density model to constrain tail allocation and an $L$-Lipschitz density model to control local concentration. The nondecreasing model gives the minimax rejection threshold, and the Lipschitz condition yields a sharper closed-form threshold with an $O(L^{-1/2})$ relative improvement over $1/\alpha$. We further show that this threshold is asymptotically sharp as $\alpha\to0$, in the sense that no uniformly valid threshold can improve on it by a fixed positive $L$-dependent amount. We then incorporate the Lipschitz calibration into multiple testing procedures with FDR control and use the same structural information to construct calibrated conditional e-values for anytime-valid sequential inference. Numerical experiments illustrate the resulting gains in discoveries and the trade-offs of the sequential procedures.

The likelihood input to a Bayesian analysis almost never exactly represents how the data were generated, calling into question the validity of posterior inferences. We review a variational interpretation of the Bayesian posterior as an alternative justification for its use under model misspecification, and consider the resulting implications on uncertainty quantification and parameter estimation. We then introduce a range of techniques that seek to obtain generalised Bayesian inferences that account for model misspecification

We study distribution testing with access to non-adaptive conditional samples. Specifically, we give tight bounds for equivalence testing, determining whether two unknown distributions are equal to or $\varepsilon$-far from each other in total variation distance. Our algorithm and lower bound show that $\tilde \Theta\left(\frac{\log n}{\varepsilon^2}\right)$ queries are necessary and sufficient for this problem. These results demonstrate that the complexity of uniformity, identity, and equivalence testing with non-adaptive conditional samples are all $\tilde \Theta(\log n)$.

Anchored Langevin dynamics (ALD) is useful for non-smooth sampling where the density of the target distribution is possibly non-differentiable and heavy-tailed; reflected anchored Langevin dynamics (RALD) can sample possibly non-differentiable target density on a constrained domain. In this paper, we propose and study non-reversible anchored Langevin dynamics (NALD) for sampling possibly non-differentiable and heavy-tailed target density in the Euclidean space and the non-reversible reflected anchored Langevin dynamics (NRALD) for sampling possibly non-differentiable target density in the constrained space. Our construction adds a circulation drift generated by a possibly state-dependent divergence-free skew-symmetric matrix field and a stream potential. It preserves the target distribution without requiring derivatives of target density, admits a random-time-change representation, and applies both on the whole Euclidean space and on bounded domains with normal reflection. By breaking reversibility, we show that NALD and NRALD can converge to their target distributions faster than their reversible counterparts via finite-time non-asymptotic convergence analysis, a large deviations analysis and asymptotic variance reduction. Numerical experiments demonstrate the efficiency of the proposed algorithms.

AI evaluation pipelines often produce conflicting judgments rather than clean labels. In pairwise LLM evaluation, this conflict is especially visible: disagreement can arise from ambiguous items, underspecified rubrics, heterogeneous or unstable human raters, or an LLM judge whose verdict changes when the response order is swapped. We propose BayesJudge, an online Bayesian meta-evaluation layer for conflicting human-LLM judgment streams. For each comparison, BayesJudge estimates a panel-relative posterior verdict distribution over the two responses, with a tie or ambiguity state when such labels are available. At the same time, it estimates rater-specific human confusion matrices and LLM presentation-order bias. The method uses tie-open labels to keep ambiguity observable and paired order-swapped judge calls to separate response quality from presentation effects. We formulate the exact online posterior recursion and use a scalable Rao-Blackwellized assumed-density SMC approximation for streaming inference. Controlled synthetic experiments demonstrate recovery of prespecified evaluator parameters and illustrate two protocol-level identifiability mechanisms: tie-open labels expose ambiguity mass, and order-swapped paired judgments separate item preference from position bias. On a real-world SummEval dataset, BayesJudge successfully detects systematic presentation-order effects in LLM judge outputs, infers distinct expert and crowdworker behavior signatures without rater metadata, and produces posterior uncertainty estimates that correlate with human disagreement. Our code is available at https://anonymous.4open.science/r/BayesJudge-0879.

The kernel bandit problem consists of sequentially optimizing an unknown function with noisy feedback, where the function has bounded norm in a given Reproducing Kernel Hilbert Space (RKHS). A central quantity in the regret analysis of kernel bandits is the maximum information gain $\gamma_T$. In particular, the best existing upper bounds scale as $\sqrt{T\gamma_T}$ up to log factors, and nearly-matching lower bounds have been derived for specific kernels such as squared exponential and Matérn. However, lower bounds for general kernels are lacking, thus making it unclear in what generality the upper bounds are near-optimal. In this paper, we establish a general $\Omega(\sqrt{T\gamma_T/\log T})$ minimax regret lower bound for non-constant continuous kernels on compact domains, establishing near-optimality (within log factors) in a very general sense. We show that the log factor appearing in this bound is unavoidable in general, but that it can be removed under certain conditions. Among other things, our findings imply that the minimax-optimal scaling is exactly $\Theta(\sqrt{T\gamma_T})$ (i.e., within constant factors) for the Matérn-$\nu$ kernel with $\nu \in (0,2)$, $\gamma$-exponential kernel with $\gamma \in (0,2)$, and certain piecewise-polynomial kernels.

We investigate the consequences of requiring a posterior approximation to preserve a specified downstream decision problem. A target posterior $P$ and loss determine a regret geometry on actions, a baseline approximation $Q_0$ determines the forward-Kullback-Leibler information required to induce action changes, and a restricted approximation family $\mathcal{Q}$ determines which such changes are available. Contracting KL divergence over Bayes-action fibers gives exact distances to decision adequacy and decision failure together with the least-informative posterior deformations that reach either side of the decision boundary. In regular finite-dimensional problems, the target and baseline constructions have quadratic local limits: a target regret Hessian $G$ and a baseline information metric $J_I$ . Their generalized eigenproblem $Gv = \gamma J_Iv$ orders local decision directions by regret consequence per unit information cost and induces a tolerance-dependent effective dimension. For restricted approximation families, the tangent image separates decision coverage from information efficiency: a family may miss consequential decision directions, or it may realize reachable directions only at excess Fisher cost. The resulting framework provides decision-relative criteria for comparing and designing posterior approximation families.

Brain signal decoding is challenging because neural recordings are noisy and vary across individuals, while labeled data are often limited. Recent attention-based models on the symmetric positive definite (SPD) manifold have nevertheless achieved strong performance using covariance and connectivity representations, yet the contribution of learned token weighting remains unclear. We examine two representative architectures, MAtt (based on log-Euclidean geometry) and GBWAtt (based on generalized Bures–Wasserstein geometry), and find that their learned attention weights remain close to uniform after training. We relate this behavior to bounded similarity parameterizations that, under the original softmax scaling, limit attention-weight contrast. Moreover, replacing learned weights with uniform weights, throughout training and evaluation, has little effect on mean predictive performance while preserving each model's original aggregation geometry. Motivated by these findings, we introduce SPD-MetaFormer, an attention-free architecture built on uniformly weighted Fréchet aggregation under log-Euclidean geometry. Its backbone uses a geodesic residual to update a summary token and a shared spectral feedforward map to transform all tokens, followed by a learned weighted readout. Token states remain SPD-valued until tangent-space classification. Across three EEG benchmarks, SPD-MetaFormer achieves competitive results relative to published Euclidean and manifold baselines. Separate matched reproductions test learned versus uniform weighting within MAtt and GBWAtt. These results suggest that, in the short-sequence and limited-data regimes studied, carefully designed SPD architectures can provide a simpler and effective alternative to adaptive manifold attention.

Real-world data often exhibit nonlinear, interactive associations between predictors and an outcome. For example, in public health studies, environmental exposures and demographic variables can have nonlinear associations with health outcomes and modify the associations of other predictors. Parametric regression models facilitate interpretable inference and hypothesis testing with Type I error rate control, but may be misspecified when associations are nonlinear. In contrast, flexible nonparametric regression models such as Gaussian processes (GPs) can capture complex nonlinearity, but are often more difficult to interpret and do not readily provide hypothesis tests with Type I error rate control. To bridge this gap, we propose a hypothesis testing framework based on additive GPs restricted to pairwise interactions, allowing tests of variable associations and interaction effects with Type I error rate control. We further establish contraction rates for additive GPs, providing theoretical support for their reduced dependence on predictor dimension when higher-order interactions are absent. Through simulations, we show that restricting the model to pairwise interactions can substantially improve testing power relative to methods based on standard GPs. Applications to environmental health studies identify nonlinear associations between metal exposures and health outcomes as well as effect modification by demographic factors.

During an infectious disease outbreak, accurate estimates of epidemiological parameters such as the reproduction number and infection duration can inform the public health response. Recently, neural simulation-based inference (NSBI) methods have become popular for estimating these parameters due to lower computational costs and more input data flexibility than traditional methods. Input data modalities that have been evaluated independently in an NSBI framework include case counts and transmission trees. Here, we evaluate whether combining information from these two data types outperforms using information from each individually with NSBI. To do so, we simulated stochastic susceptible-infectious-recovered (SIR) outbreaks across a range of basic reproduction numbers (R0) and average infection durations (D) to generate training, validation, and testing datasets. We then calculated, for each simulated outbreak, summary statistics from case count trajectories and from transmission trees. Using these features, individually and in combination, we performed a type of NSBI called neural ratio estimation (NRE) to estimate R0 and D. We also calculated maximum likelihood estimates of the parameters by fitting a deterministic SIR model using the case count trajectory. All three NRE models outperformed the baseline method. Furthermore, incorporating tree summary statistics improved the estimation of both R0 and D over case count summaries alone. This suggests that trees provide valuable additional information for estimating epidemiological parameters above and beyond case counts alone.

One-to-one matching without replacement is a classical approach to constructing comparable treated and control samples in the design of observational studies. It pairs each treated unit with a distinct control while minimizing a covariate distance objective. With continuous covariates, the matched pairs generally remain inexact, which contributes to bias in downstream analysis. Its key theoretical properties, such as the resulting imbalance between matched pairs and when it is negligible to support valid inference, remain unclear. In this paper, we analyze one-to-one matching based on $d$-dimensional, continuous covariates with a quadratic covariate-distance objective. First, we find that when $d\leq 3$, under standard conditions on the propensity score ensuring abundant control samples near each treated sample, the imbalance (difference between within-group averages) is root-$n$ negligible uniformly over the family of smooth functions with a common first- and second-order derivative bound. However, such balance is subject to a dimension restriction, as we construct examples in which the imbalance is root-$n$ non-negligible when $d=4$ and dominates root-$n$ rate when $d>4$. Second, we show that when $d\leq 3$, the matched design allows valid Wald-type and bootstrap inference for the average treatment effect on the treated, distributional treatment effects, and quantile treatment effects. Thus, the same outcome-blind matched design supports various downstream inferences without having to tailor the design to the targets. Finally, paired randomization inference based on the matched design is asymptotically valid in the super-population sense for $d\leq 3$ but can fail when $d=4$. We corroborate the theoretical results with numerical experiments.

Markov chain Monte Carlo (MCMC) methods are the standard tool for computing expectations under complex probability distributions. Control variates reduce the variance of the resulting estimates, but a good control variate requires solving the Poisson equation of the sampler, which rarely admits a closed-form solution. Exact solutions are available when the sampler's kernel has a known spectral decomposition on a simple reference density. In our work, we extend these solutions to general targets through a learned change of variables. A bijection, such as a normalizing flow, is trained so that the target becomes close to the reference in a latent space, and we show that Markov kernels and their Poisson solutions are transformed by any bijection. Running such samplers in the latent space then yields explicit control variates, and the estimator is consistent under mild tail conditions on the map and target. Importance sampling (IS) from the flow is the limiting case of the same construction and the control variates apply to it as well. Experiments on synthetic targets and real posteriors compare the procedure against state-of-the-art samplers and control variates.

We establish nearly-matching converse bounds on the sample complexity of estimating the average mixing time of a Markov chain from a single stationary trajectory. Unlike the worst-case mixing time, which maximises the total-variation distance to stationarity over all initial states, the average mixing time averages this distance under the stationary distribution and can therefore be substantially smaller. Our converse bounds establish when estimation is impossible, and match the key dependencies in existing achievability bounds, up to logarithmic factors. Together, these bounds provide nearly-tight PAC guarantees for estimating the average mixing time.

Many treatments have effects that plausibly vary by latent classes. For example, the effect of personalized video recommendations may be stronger for users who seek a new discovery than for users seeking to rewatch something familiar. In the context of user journeys on digital platforms, we can conceptualize user moods as dynamic latent states that lead to distinct action distributions. Because mood is unobserved, we infer it from a user's pre-treatment actions using a hidden Markov model (HMM), and then leverage the latent state as an interpretable treatment effect modifier when evaluating experiment results. Under a treatment-effect sufficiency assumption, each observable conditional effect is a known mixture of the latent effects, so state-specific effects are identified when state posteriors are sufficiently heterogeneous. We propose an estimator that is Neyman-orthogonal to errors in the fitted HMM, enabling standard influence-function confidence intervals. In a real-world large-scale A/B test of a new recommendation model at Netflix, our latent-state analysis reveals that heavy-engagement users show significantly positive effects across all treatment variants, with limited evidence of benefit among less-active and rewatch-oriented users.

Optimal dynamic treatment regimes (ODTRs) are increasingly used to generate sequences of personalized recommendations that maximize a final outcome of interest. Although ODTRs are widely used in biostatistics and precision medicine, their applications in education remain limited, especially for longitudinal decision-making. This study translates ODTR methods for single- and multi-stage designs into individualized high school math course recommendations, providing step-by-step guidelines and demonstrations with data from the High School Longitudinal Study of 2009 (HSLS:09). We use Targeted Maximum Likelihood Estimation with ensemble machine learning to estimate ODTRs that maximize students' math achievement and college enrollment, imposing feasibility constraints based on domain knowledge and propensity-score thresholds to ensure recommendations are practically implementable. Results show that unconstrained regimes yield higher estimated values but infeasible recommendations, while feasible regimes produce realistic pathways with stronger empirical and expert-informed support. Finally, we discuss practical considerations for designing ODTRs in education.