Papers

Watermarking has been proposed as a way to identify synthetic samples in estimation settings where no metadata is available to distinguish them from real samples, but its precise effects remain unexplored. In the absence of a distinguishing mechanism, it has been shown that adding synthetic samples significantly reduces the marginal efficacy of new real samples. In this work, we study the minimax loss of such recursive discrete distribution estimation in the presence of watermarks in contrast to the unassisted and oracle-assisted losses. When the fraction of real samples vanishes asymptotically, we provide a lower bound that shows that it is impossible to improve performance by adding watermarks unless the false negative rate of detection also vanishes. Additionally, we show that in most regimes, the worst-case losses of a sequence of simple deterministic estimators match the corresponding lower bounds up to constants. Finally, we propose masking, a randomization procedure that narrows the gap in the remaining regimes to a Jensen gap. We conjecture that a tighter lower bound argument can close this gap.

Automatic vehicle location (AVL) data produced by transit vehicles is invaluable in performance studies, but turning raw AVL points into a detailed view of vehicle stop-and-gos is burdensome: the datasets are sparse, noisy, and prone to blunders. While recent research has explored methods of reconstructing trajectories describing the position of vehicles over time, the common techniques can be complex, and no open-source tools exist to help practitioners process the raw AVL. We fill this gap by proposing a thorough methodology for cleaning transit AVL data and providing an open-source R package, built on open data standards, to implement the workflow and reconstruct vehicle trajectories, allowing practitioners to easily formulate custom microscopic performance metrics. Using a large AVL dataset with over 3,000 trips from Indianapolis, Indiana, we demonstrate the workflow, evaluate the package's efficiency, and demonstrate the utility of the trajectories by estimating various traffic signal performance metrics. Finally, we use cross-validation to quantify the error in reconstructed trajectories at various polling frequencies. We find that the proposed data processing methodology and tool are efficient, requiring roughly 3 minutes of processing time on the large dataset, and error in position estimates is low (root mean square error under 10 meters at polling frequencies of 15 seconds). The estimated signal performance metrics can inform future evaluations of signal programming along the corridor. In sum, this paper serves as a practical guide and complementary toolbox for practitioners seeking to use transit AVL data to build a detailed view of vehicle stop-and-go cycles.

When do text embeddings work as inputs to empirical analysis? Their use rests on an assumption: that we can trade text for its low-dimensional embedding, and lose little in doing so. I make that assumption precise under a generative model in which documents are mixtures of latent topics. I study two uses—clustering units in embedding space and controlling for high-dimensional text. A cluster of embeddings is a set of documents with similar topic mixtures; controlling for the embedding is equivalent to controlling for the topic mixture, so validity reduces to whether that mixture captures the confounding. In an application to 363 U.S. metropolitan areas, embedding-based clusters of LLM-generated economic descriptions recover interpretable economic archetypes and separate local employment dynamics more sharply than clustering on model residuals, or on a curated set of industry and demographic covariates.

Calibration under model misspecification is inherently ill-posed because calibration parameters and structural discrepancy are statistically confounded without additional assumptions. Bayesian formulations address this ambiguity through prior and covariance modeling choices, including multi-response observations and cross-source alignment. However, how these choices, together with output-dependent uncertainty, shape posterior geometry and practical parameter identifiability remains poorly understood. We present a latent-variable multi-response calibration model that combines empirical Bayes estimation of nuisance hyperparameters with a Fisher-information-based local Gaussian approximation of the calibration-parameter posterior. Closed-form expressions are derived for squared-exponential and Matérn covariance functions, revealing how multi-response correlations, latent alignment, and output-specific uncertainty influence local posterior geometry. Controlled numerical experiments show that additional responses provide complementary geometric constraints on calibration parameters, while noise modeling can improve uncertainty quantification by reducing overconfidence without necessarily improving parameter localization. A lithium-ion battery study further illustrates how the model distinguishes strongly from weakly constrained parameters. Together, these results establish expected-Fisher posterior geometry as a scalable diagnostic of local parameter identifiability in multi-response calibration.

Scientific discovery often requires reasoning over competing hypotheses that are consistent with experimental observations. For mixed-variable and combinatorial hypothesis spaces, however, constructing probabilistic representations remains challenging because both the active model components and their associated parameters are unknown. In this work, we present a framework for learning continuous latent representations of admissible partial differential equations (PDEs) by embedding a scientific inductive bias directly into the training distribution. Progressively richer structural principles (e.g., sparsity, logical dependencies, common PDE families, and physical admissibility) are used to generate a structured distribution of hypotheses from which a gated variational autoencoder learns a continuous latent manifold. Experimental results show that the resulting 11-dimensional representation accurately reconstructs a broad collection of representative PDEs, while exhibiting smooth geometric transitions both within and across equation families. Through an ablation study we further demonstrate that introducing scientific principles reduces both structural misclassifications of equation forms and parameter estimation errors when reconstructing a representative benchmark set of admissible partial differential equations. These results show that embedding a scientific inductive bias in the training distribution enables the learning of compact and geometrically meaningful hypothesis manifolds, providing a principled foundation for future inference over competing governing equations.

Agent evaluations often use one benchmark to choose a workflow and then search for task types where its advantage weakens, so both conclusions are selected from the same data. We introduce Selection-Aware Semantic Stress Testing (\SASST{}), which learns a task reweighting from pre-execution features on discovery tasks and evaluates the same paired comparison on separate confirmation tasks. The protocol checks support and stability, uses joint bounds for all planned claims, and can return no claim. We prove conditional asymptotic validity under stated cluster assumptions. A forty-cluster audit finds Gaussian undercoverage and conservative Bonferroni $t$ bounds. In one 480-episode $\tau$-bench study, a $3.75$ point discovery gain vanished on confirmation. A second-model study likewise confirmed neither a workflow benefit nor a stable stress rule.

Extreme and rare changes in space mission solar irradiance records are scientifically relevant but difficult to quantify because these records are finite, instrument dependent, and affected by observational gaps and time varying measurement quality. We evaluated extreme daily logarithmic changes in the band integrated 0.1-7.0 nm irradiance measured by photodiode 7 of the Solar Radiation and Climate Experiment/X-Ray Photometer System (SORCE/XPS) from 2005 to 2019. After constructing a regular daily series by linear interpolation, we analyzed daily logarithmic changes in irradiance and fitted stationary Generalized Extreme Value models to 60 day block maxima and transformed 15 day block minima. Block lengths were selected using Ljung-Box diagnostics and sample autocorrelation functions. Maximum likelihood estimation was used as the primary inferential method, with probability weighted moments as a sensitivity check. The maximum likelihood GEV shape estimates were 0.1661 for maxima and 0.2928 for transformed minima, with closely aligned estimates under the two fitting methods. These positive point estimates are compatible with Fréchet type tails under the selected block constructions. Annualized return levels provide interpretable summaries of extreme relative increases and reductions, but estimates for long return periods remain strongly dependent on extrapolation beyond the 15 year record. Reported measurement precision and absolute uncertainty were used to qualify the interpretation of the fitted tails and were not propagated through the likelihood. By combining EVT based tail modeling with explicit consideration of measurement precision, absolute uncertainty, interpolation, and mission data gaps, the analysis provides an uncertainty aware astrostatistical baseline for extreme value inference from processed solar mission records.

Constructing confidence regions for stochastic gradient descent (SGD) ideally requires estimating the asymptotic covariance matrix, a severe computational bottleneck in high dimensions. Traditional cancellation-based batch means methods bypass this estimation but require inverting a sample batch covariance matrix. This introduces strict mathematical degeneracy when the parameter dimension exceeds the number of batches. To address this problem, we utilize equal batch size batch means method and propose a simultaneous, marginal-friendly framework. The proposed marginal statistics has a asymptotic Student's $t$-distribution, and eliminates the matrix inversion step, entirely circumventing high-dimensional degeneracy. To achieve valid simultaneous coverage, we present an algorithm utilizing wild bootstrap samples drawn from a statistic as a function of only the diagonals of the variance-covariance estimator, and to further incorporate the contribution of cross-dependencies, we introduce an efficient Quasi-Monte Carlo procedure utilizing a $t$-copula approximation. Additionally, we integrate a Lugsail variance estimator to aggressively correct finite-sample bias and under-coverage. The proposed methodology delivers interpretable, simultaneous hyper-rectangular confidence regions that are statistically robust, memory-efficient, and strictly scalable for high-dimensional inference. The theoretical results are supported by extensive numerical simulation analysis through various aspects of dimension, number of batches and error structure.

Natural-language tasks can elicit different verdicts from protocol-following evaluators that receive the same declared information. We study aggregate disambiguation systems (ADSs). Given a task and a candidate solution, each evaluator casts a binary vote on whether the solution should be accepted, and the system aggregates the votes of a finite panel. The target is protocol reproducibility relative to an explicitly declared evaluator reference, not semantic truth. We separate fixed finite censuses, probabilistic evaluator populations, and growing-census limits, since their endpoint laws and guarantees are not interchangeable. In the population setting, we use finite samples to estimate how often a finite panel reaches the same decision as the declared evaluator population. We provide a lower confidence bound on the fraction of candidate solutions for which the disagreement probability is at most a chosen tolerance. The calculation accounts separately for sampling candidate solutions and sampling evaluators. The construction permits arbitrary dependence among columns induced by shared evaluator rows and uses exact binomial intervals at the evaluator layer and an exact one-sided binomial inversion at the generator layer. Simulations check the implementation against known population coverages and expose power limitations.

Clinical validation studies of predictive tests are usually designed to focus on sensitivity ($Se$) and specificity ($Sp$), while statistical power is often calculated on regression-effect scales (e.g., risk ratio, hazard ratio). However, these quantities are statistically connected. Here, we provide closed-form links from sensitivity, specificity, and disease prevalence ($\pi$) to predictive risks, risk contrasts, and Wald-type variance, power, and sample-size formulas for binary and fixed-horizon survival outcomes. Analyses of statistical efficiency via C- and D-optimal principles demonstrate how prevalence and threshold choices affect study efficiency, supporting rapid decisions in preliminary studies and informing the design of subsequent, larger studies. Simulations show good calibration across most realistic scenarios; when events are rare and test effects are simultaneously very large, continuity and minimum-event corrections are needed to stabilize the approximation. We illustrate the framework with a case study describing use of the coronary artery calcium score for predicting incident cardiovascular disease in patients with type 2 diabetes mellitus. The formulas let investigators check power and required enrollment directly from $(Se,Sp,\pi)$, without running a separate simulation for each design candidate.

For two Gaussian random elements on a separable Hilbert space with common covariance operator, the fourth-order moment tensor structure of their mixture admits an operator-valued representation obtained from the purely quadratic component of the class-conditional second moment. We identify the tensor mechanism generating the completely diagonal coefficients of this fourth-order representation after covariance standardization, and show that these coefficients are governed entirely by the coordinatewise Cameron-Martin energy of the mean shift. This yields a necessary and sufficient condition, together with an explicit eigenvalue, for the Fisher discriminant to be an eigenfunction of the induced coordinatewise operator. We further introduce aggregated fourth-order spectral functionals for two complementary constructions, one built from fourth-order moments and the other from a product-of-expectations counterpart, and prove that they are asymptotically equivalent precisely when the underlying Gaussian measures are mutually singular. These results provide a spectral realization of the classical Cameron-Martin criterion through higher-order moment operators, explaining the probabilistic origin of fourth-order spectral quantities previously proposed for Gaussian discrimination and functional data classification and relating them to the "near-perfect classification" regime.

We develop a marginal coordinate test for regression with Euclidean predictors and a random-object response in a separable metric space. The goal is to test whether a predictor provides additional information about the response conditional on the remaining predictors. In a semi-supervised design, an unlabeled sample is used to estimate predictor conditional means, while an independent labeled sample is reserved for inference. The resulting residuals are combined with a product-space kernel to form a kernel conditional mean dependence (KCMD) U-statistic without requiring a response residual. The primary identity-based test targets a necessary conditional mean restriction, while a multiple-transformation extension probes broader alternatives. We establish a weighted centered chi-square null limit, wild bootstrap validity, consistency against fixed detectable alternatives, and local power under mean-element alternatives. For simultaneous inference, truncated p-to-e calibration combined with e-BH provides asymptotic false discovery rate control under general dependence. Simulations with Euclidean and non-Euclidean responses, together with a New York City taxi-flow analysis, illustrate the method.

Agricultural commodity prices often exhibit strong temporal persistence, which may limit the performance of conventional time series models. This study investigates long memory in logarithmic monthly white maize prices from six major markets in Togo between January 2001 and June 2022. Long memory is examined using the Geweke–Porter–Hudak, Local Whittle, Exact Local Whittle, and wavelet log-regression estimators. SARIMA, ARFIMA, and SARFIMA models are subsequently compared using the Bayesian Information Criterion and residual diagnostics. Long-range dependence is found across all markets. Fractionally integrated models provide the best fit for most markets, although SARIMA remains preferable for some. The results demonstrate that evidence of long memory does not necessarily imply that a fractionally integrated model provides the best empirical fit, emphasizing the importance of data-driven model selection.

Informative label missingness can change the usual efficiency ordering between completely and partially labelled classifiers because the pattern of missing labels may itself carry information about the classification model. We develop a general likelihood-based theory for this phenomenon in parametric multiclass classification. An efficient-information decomposition separates information lost through unavailable class memberships from information contributed by the missing-label mechanism. We then derive a quadratic expansion of plug-in excess risk over the active pairwise faces of the multiclass Bayes boundary, showing that classification efficiency depends on how information gains and losses align with directions that perturb the decision boundary. This yields a classification-weighted generalized-eigenvalue criterion under which informative partial classification may have smaller asymptotic classification risk without globally dominating complete classification in Fisher information. Near missing completely at random, with the marginal missing-label proportion fixed, redistribution of missing labels changes lost class-label information at first order, whereas efficient information from the missingness pattern appears only at second order. Three-class quadratic discriminant calculations, finite-sample experiments, and a semi-synthetic multiclass application illustrate the resulting regime-dependent behaviour.

Machine learning systems are increasingly corrected while they run, and the decision of when to intervene is increasingly delegated to statistical monitors. Anytime-valid inference promises evidence that can be acted on at any moment, exactly the guarantee this setting needs, and it is moving from theory into deployed monitoring. Conformal test martingales are the change-detection instrument, and Ville's inequality caps their false-alarm probability on exchangeable data. The guarantee is conditional. A deployment inherits it only if the stream it monitors behaves exchangeably. The premise is hardest to satisfy where these monitors are most useful, on dependent data and inside loops where the monitor modifies the learner whose scores it reads. It is also rarely measured. We measure it in a pre-specified case study, where such a monitor gates the online updates of a Kalman adapter correcting frozen time-series foundation models on five forecasting streams. On exchangeable synthetic streams, the same implementation fires in at most 1 of 60 runs. On the real streams, at alpha = 0.05, 135 of 135 clean-stream runs fired. The construction does not explain the firing; the failure comes from the deployed score stream itself. Repeated fires hold the gate's drift response active, and the gated filter amplifies the very transient it was designed to prevent. The component worth keeping makes no validity claim. Huber-style gating of the filter's own updates cuts isolated-spike degradation by an order of magnitude with no dataset specific tuning. Anytime-valid methods proposed for dependent data should therefore be accompanied by null-calibration controls and mechanism traces.

Differential privacy has become the standard for generating privacy-protected data releases. However, differential privacy does not translate intuitively to disclosure risk. In particular, it remains unclear how much an adversary's belief about an individual's inclusion in a dataset can change after observing a protected release. To address this question, we derive upper and lower bounds on the posterior-to-prior ratios of inclusion beliefs under bounded probabilistic and approximate differential privacy. By assuming a worst-case adversary with all-but-one auxiliary information, i.e., knowledge of all except for one of the participants in a dataset, we obtain bounds that apply to any adversary. Because these bounds may fail with non-zero probability, we study the corresponding failure probability for the Gaussian mechanism. We derive a theoretical upper limit on this probability and compare it with Monte Carlo estimates across a wide range of parameter settings. The observed failure rate is several orders of magnitude smaller than its theoretical upper limit, indicating that the latter is highly conservative. These findings suggest that the inferential privacy guarantees provided by differentially private mechanisms may be substantially stronger in practice than what is implied by the theoretical upper limit.

Recent offline reinforcement learning (RL) studies report policies that outperform physician decisions on clinical outcomes. We conduct a systematic, partially crossed evaluation of five offline RL algorithm families and 14 reward designs in 44,894 post-2018 acute ischemic stroke patients from a nationwide registry (N = 129,033). Standard Fitted Q-Evaluation (FQE) yields an apparent policy-improvement estimate of +0.0069; adding an Early Neurological Deterioration penalty increases it to +0.0101. We identify reward-embedded confounding, in which a proxy terminal reward encodes baseline severity and prognosis as well as treatment efficacy. A 2 x 2 factorial analysis finds that terminal reward confounding accounts for 218.6% of the observed signal change, so its removal overshoots the null. After DML-inspired GBM reward residualization, the FQE estimate attenuates to +0.0033 (p = 0.132), and full deconfounding yields +0.0025 (p = 0.291). FQE-based diagnostics, T-learner analyses, and direct recurrence analyses converge away from a clinically meaningful aggregate improvement. A 1-year mRS factorial analysis replicates the attenuation. We provide an empirically motivated six-step evaluation checklist. NIHSS-stratified heterogeneity is hypothesis-generating for prospective trial design; hospital-level disagreement does not persist after full reward deconfounding.

Mediation analyses are increasingly conducted in randomized trials, but a sample size adequate for the total treatment effect may leave the natural indirect effect (NIE) or natural direct effect (NDE) substantially underpowered. Randomization does not extend to the mediator, so precision depends on the conditional mediator distribution and the mediator-outcome association, neither of which enters a total-effect calculation. Planning outside linear structural equation models is largely based on simulation under a fully specified data-generating mechanism rarely available at the design stage. This paper develops analytic power and sample size formulas for the NIE and NDE with a binary mediator and a continuous or binary outcome. Under standard identification assumptions, we focus on the ratio-of-mediator-probability weighting (RMPW) estimator that does not require an outcome model for effect estimation. We decompose the oracle variances of the RMPW estimators into components capturing mediator-probability-ratio variability, outcome variation, and their association, with an additional shared-arm covariance term for the NIE. Under a probit latent-index mediator and a working outcome model, these components are determined by a small number of interpretable design inputs rather than by the full joint distribution of covariates, mediator, and outcome. Simulations show that the analytic sample sizes closely match simulation-based benchmarks, attain the target power, and maintain type I error near the nominal level. An ACTG175 illustration shows how pilot data can calibrate the inputs.

We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with \(\tau\geq m\), we prove that the Wishart splitting matrix \(W\) gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous \(99\%\) Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal–ranked–equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.

By slightly adapting Wang's recent important argument for the homoscedastic setting, we prove that every finite heteroscedastic Gaussian mixture has only finitely many critical points, and, consequently, only finitely many modes.

Bergsma–Dassios sign covariance $\tau^*$ is a rank-based population measure of dependence. Building on zero-characterisation results under specific regularity regimes, we prove that $\tau^*(X,Y)=0$ characterises independence for every real-valued bivariate distribution, including mixed and singular laws. For the unnormalised four-sample convention for $\tau^*$ defined in Subsection 4.3 and the unscaled Blum–Kiefer–Rosenblatt functional $\mathscr {B}$, the proof gives the quantitative inequality $\tau^*\ge 2\mathscr {B}$. This is a population identification result; no new sample-level limit theorem is claimed. The argument first encodes finite ordered distributions with rational cell probabilities by labelled path trees and applies a nonnegative sum-of-squares representation for a quartet covariance. Rational approximation and nested quantisation then remove all support and regularity restrictions. On finite uniformly weighted label sets, the tree framework also relates an edge-weighted quartet quantity to empirical distance covariance squared. As a separate combinatorial consequence, it yields the asymptotic $2/3$ upper bound for the quartet distance between binary phylogenetic trees.

Classical numerical solvers for partial differential equations (PDEs) are computationally expensive to solve repeatedly across varying initial conditions, motivating the need for learned surrogates. In this paper, we propose a trainable Neural Cellular Automata (NCA) based surrogate model for learning long time PDE dynamics. Rather than mapping an entire initial field to a full trajectory in one shot, our proposed model learns a small, local, homogeneous update rule that is applied identically and repeatedly at every grid cell, mirroring the locality of differential operators. We benchmark this framework against three baselines: PDE - Net, a modified physics-informed neural network (PINN), and a Fourier Neural Operator (FNO), on five canonical PDEs (heat, advection, Burgers, Allen - Cahn, and Fisher - KPP), evaluated at temporal domain two times beyond the training temporal domain. The proposed model achieves the lowest long-horizon relative errors on the majority of the experiments.

We estimate the conditional population-risk curve of a realized smooth nonconvex gradient flow from the training sample. Flow approximate leave-one-out (Flow-ALO) propagates a deletion response and evaluates omitted observations at approximate deleted paths. The risk-curve error decomposes into response approximation, exact-LOO fluctuation, and deletion-to-full risk transfer. On each fixed finite horizon, bounded centered training-loss gradients, a one-sided Hessian lower bound, locally Lipschitz Hessians, and a strict tube-closure condition yield an explicit $(n-1)^{-2}$ bound for the deletion-response error. Bounded evaluation-loss gradients transfer the deletion-response bound to the score without requiring the Hessian to be invertible. Direct first-order jackknife cancellation and exact-LOO concentration control deletion-to-full risk transfer and fluctuation, respectively, completing recovery of the conditional population-risk curve. For bounded smooth two-layer mean-field networks training both layers, the score-error bound is uniform in width.

We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. We study vector-valued linear regression with square loss $\ell_{(x,y)}(W)=\lVert Wx-y\rVert_2^2$, where $x\in\mathbb{R}^d$ and $y\in\mathbb{R}^m$. The learner returns the minimum-Frobenius-norm empirical risk minimizer. We prove that the minimal budget of weighted examples for recovering the full-data loss on every finite dataset is exactly $n^{\star}(d,m)=(m+1)d$. We determine the weighted selection profile $F_{\mathrm{weighted}}(d,m,n)$ at the near-threshold budget: $F_{\mathrm{weighted}}(d,m,(m+1)d-1)=1+1/(dm^2)$. We recover the known spanning-budget value $F_{\mathrm{weighted}}(d,m,d)=d+1$ for every $m$, and $F_{\mathrm{weighted}}(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_{\mathrm{weighted}}(2,2,3)\in[13/8,15/8]$ and $F_{\mathrm{weighted}}(2,2,4)\in[5/4,3/2]$. We reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms and assemble structural evidence for it. The upper bounds use a fixed-basis conic compression lemma, a determinant–facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems. These tools may be of independent interest. We also exhibit an explicit six-point integer dataset with $d=m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. Thus the scalar sufficient budget $2d$ does not extend to vector-valued outputs. Our new regression-profile results for $m\ge2$ extend the scalar theory for $m=1$.

Markov chain pool decoding (MCPD) devised by Knill et al. (1996) identifies likely positive clones from noisy pooled-test results. The standard MCPD estimates clone-wise posterior probabilities using Gibbs sampling, but it may allocate excessive computational effort to low-scoring clones. This paper focuses on parallelizing MCPD on GPU architectures. Whereas the standard MCPD employs systematic-scan updates, we propose a score-weighted update scheme that updates high-scoring clones more frequently. We prove that the stationary distribution of the proposed Markov chain coincides with the target posterior distribution. To enable efficient GPU parallelization, we further incorporate the unbiased MCMC framework of Jacob et al. (2020) and employ a slot-refilling technique based on the arguments by Glynn and Heidelberger (1991) about the coupling of Markov chains. Experiments involving 1,298 clones, 97 pools, and three true positives demonstrate improved recovery compared with uniform decoders, while maintaining high overlap under high-noise conditions.

We propose a new design of fair classifiers for multi-class classification problems in the presence of vector-valued sensitive attributes. In that scenario each sensitive attribute has multiple values and forms several groups relevant to the fairness consideration. Naturally those groups are overlapping and one should also analyze the interaction of factors. Additionally, the decision makers aided by the classification should not violate individual rights at the expense of satisfying fairness metrics at the group level. We propose an approach using the theory and methods of coherent measures of risk aiming at resolving the fairness challenges. Further, we propose a specialized numerical method for solving the resulting optimization problem. The method scales well with the increase of the number of observations. Additionally, we note that the obtained classifier is robust with respect to corrupted data or to situation when data is scarce. We demonstrate the advantages of the proposed framework in comparison to the support-vector machine framework and other methods handling fairness.

For Gaussian approximation over high-dimensional rectangles under unrestricted covariance, Chernozhukov et al. (2023b) conjectured that the $n^{-1/4}$ rate, up to logarithmic factors, is near-optimal. We show that, under the coordinatewise subexponential condition with scale $B_n$ and the marginal variance lower bound condition with constant $b$ in Chernozhukov et al. (2023b), the approximation error in dimension $d$ is bounded by \begin{align*} C_b\min\left\{ 1,\, \left(\frac{B_n^2}{n}\right)^{1/3}\{\log(2dn)\}^{7/3} + \frac{B_n}{\sqrt n}\{\log(2dn)\}^{5/2} \right\}. \end{align*} In particular, for bounded $B_n$ and polynomial dimension, the new bound is $n^{-1/3}$ and therefore falsifies the polynomial-dimensional $n^{-1/4}$ near-optimality conjecture. The proof uses a two-stage interpolation and a rank-free matrix-weighted Gaussian surface bound, which may be of independent interest. The initial proof attempt was generated by ChatGPT 5.6 Pro (OpenAI) and subsequently corrected and rewritten by the authors. The machine-checked Lean formalization of the proof can be found at the GitHub repository (https://github.com/WeihanZhang2001/cubic-root-gaussian-approximation-under-unrestricted-covariance).

Studies with staggered entry, in which individuals enroll at different calendar times, are ubiquitous in medicine and related disciplines. Because these studies usually have a fixed administrative end of follow-up, identification of the estimand of interest relies on assumptions about the right-censoring mechanism. The assumptions are often considered plausible only conditional on covariates, including the time an individual entered the study (E). Yet, censoring assumptions formulated conditional on E are ill-posed due to positivity violations. Here, we study the consequences of this issue for common procedures in causal survival analysis, such as those based on marginal structural models. We further give conditions for valid identification and introduce sensitivity analyses to assess practical implications. We illustrate our methodology through two case studies. The first builds on the seminal article by Hernán et al. (2000) on the effect of zidovudine treatment. The second reanalyzes a recent study on the effect of the mRNA vaccine in patients receiving immune checkpoint inhibitors.

We develop a multifidelity method for fusing low-resolution simulations with computationally expensive high-resolution simulations, which are run infrequently and are therefore prone to bias. We formulate this fusion as a constrained optimization under missing-not-at-random (MNAR) selection bias. This formulation searches for the exponentially tilted high-resolution distribution that minimizes KL divergence from the biased baseline, subject to moment constraints derived from low-resolution simulations. This optimization requires first estimating the biased baseline conditional density $f$ as a nuisance parameter. We estimate $f$ using a score-based diffusion model. To eliminate the generative model's regularization bias that harms the downstream task, we apply targeted maximum likelihood estimation (TMLE). TMLE debiases $\hat{f}$ via a targeted exponential tilting, rendering the target parameters insensitive to first-order nuisance estimation errors. To execute this computationally, we adapt generative model steering, a technique originally developed for human-preference alignment. Using Feynman-Kac steering with a reward function based on our formulation, we simultaneously execute the exponential tilts for MNAR and TMLE at inference time, avoiding expensive retraining costs. Code available [here](https://github.com/Jong-Min-Moon/multifidel_emul_by_FK).

Researchers often summarize mediation in clustered data with a single-level product-of-coefficients estimator. We show what this pooled analysis estimates and why the target depends on the design. The pooled a-path is weighted by the within-between variance composition of the treatment. The pooled b-path uses a different weight, based on the mediator variance that remains after residualizing on treatment and covariates. Because these weights generally differ, the pooled indirect effect can differ from a level-respecting combination of the within- and between-level indirect effects. We derive the pooled probability limit and split this difference into a nonlinearity term and a weight-mismatch term. Equal b-paths remove the distortion, but equal a-paths do not. Allowing the two weights to vary gives a bilinear surface with saddle geometry, sharp corner bounds, and a zero-distortion frontier. We also define a design-standardized indirect effect and practical diagnostics. Simulations verify the identities and show that a common decompose-first, cluster-weighted construction can overstate apparent distortion. In an ECLS-K:2011 illustration with 8,363 children in 753 baseline schools, the within-level weights were 0.723 and 0.896, a difference of -0.173 (95% bootstrap CI [-0.193, -0.151]). The conventional construction produced 1.76 times the absolute distortion of the identity-consistent construction. The identity-consistent AD interval included zero. Thus, the application shows unequal weighting and sensitivity to construction, not evidence of nonzero population distortion.