Papers

The concept of ranking aggregation plays a central role in preference analysis, and numerous algorithms for calculating median rankings, often originating in social choice theory, have been documented in the literature, offering theoretical guarantees in a centralized setting, i.e., when all the ranking data to be aggregated can be brought together in a single computing unit. For many technologies (e.g. peer-to-peer networks, IoT, multi-agent systems), extending the ability to calculate consensus rankings with guarantees of convergence and resilience to potential contamination in a decentralized setting, when preference data is initially distributed across a communicating network, remains a major methodological challenge. Indeed, in recent years, the literature on decentralized computation has mainly focused on computing or optimizing statistics such as arithmetic means using gossip algorithms. The purpose of this article is precisely to study how to achieve reliable and resilient consensus on collective rankings in a decentralized setting, thereby raising new questions, robustness to corrupted nodes, and scalability through reduced communication costs in particular. The approach proposed and analyzed here relies on the robustness guarantees offered by random gossip communication, which allows autonomous agents to compute a global ranking consensus using local interactions only, without coordination or a central authority.

We are honoured to have our work read and discussed at such a thorough level by several experts. Words of appreciation and encouragement are gratefully received, while the many supplementary comments, thoughtful reminders, new perspectives and additional themes raised are warmly welcomed and deeply appreciated. Our thanks go also to JASA Editor Francisco Samaniego and his editorial helpers for organising this discussion. Space does not allow us answering all of the many worthwhile points raised by our discussants, but in the following we make an attempt to respond to what we perceive of as being the major issues. Our responses are organised by themes rather than by discussants. We shall refer to our two articles as `the FMA paper' (Hjort and Claeskens) and `the FIC paper' (Claeskens and Hjort).

Multi-armed bandit (MAB) processes constitute a foundational subclass of reinforcement learning problems and represent a central topic in statistical decision theory. Yet, conducting valid sequential testing under adaptive allocation remains challenging due to the lack of asymptotic theory under non-i.i.d. reward sequences and sublinear sample sizes for some arms. To address this open challenge, we propose an Urn Bandit (UNB) process to integrate the reinforcement mechanism of urn probabilistic models with MAB principles, ensuring almost sure concentration of allocation proportions on optimal arms. We establish a joint functional central limit theorem (FCLT) for consistent estimators of expected rewards under non-i.i.d. reward sequences with non-sub-Gaussian tails and pairwise cross-arm dependence. To overcome the limitations of existing methods that focus mainly on cumulative regret and therefore provide only algorithmic performance guarantees without supporting valid sequential testing, we develop an asymptotic theory for sequential test statistics under the proposed UNB process. The resulting framework enables a broad class of sequential inference procedures, such as A/B testing and policy evaluation. Simulation studies and real data analysis demonstrate that UNB maintains testing performance comparable to that of the equal randomization (ER) design while achieving improved reward accumulation relative to ER.

We decompose physician disagreement in the HealthBench medical AI evaluation dataset to understand where variance resides and what observable features can explain it. Rubric identity accounts for 15.8% of met/not-met label variance but only 3.6-6.9% of disagreement variance; physician identity accounts for just 2.4%. The dominant 81.8% case-level residual is not reduced by HealthBench's metadata labels (z = -0.22, p = 0.83), normative rubric language (pseudo R^2 = 1.2%), medical specialty (0/300 Tukey pairs significant), surface-feature triage (AUC = 0.58), or embeddings (AUC = 0.485). Disagreement follows an inverted-U with completion quality (AUC = 0.689), confirming physicians agree on clearly good or bad outputs but split on borderline cases. Physician-validated uncertainty categories reveal that reducible uncertainty (missing context, ambiguous phrasing) more than doubles disagreement odds (OR = 2.55, p < 10^(-24)), while irreducible uncertainty (genuine medical ambiguity) has no effect (OR = 1.01, p = 0.90), though even the former explains only ~3% of total variance. The agreement ceiling in medical AI evaluation is thus largely structural, but the reducible/irreducible dissociation suggests that closing information gaps in evaluation scenarios could lower disagreement where inherent clinical ambiguity does not, pointing toward actionable evaluation design improvements.

Aggregation constraints, arising from geographical or sectoral division, frequently emerge in a large set of time series. Coherent forecasts of these constrained series are anticipated to conform to their hierarchical structure organized by the aggregation rules. To enhance its resilience against potential irregular series, we explore the robust reconciliation process for hierarchical time series (HTS) forecasting. We incorporate M-estimation to obtain the reconciled forecasts by minimizing a robust loss function of transforming a group of base forecasts subject to the aggregation constraints. The related minimization procedure is developed and implemented through a modified Newton-Raphson algorithm via local quadratic approximation. Extensive numerical experiments are carried out to evaluate the performance of the proposed method, and the results suggest its feasibility in handling numerous abnormal cases (for instance, series with non-normal errors). The proposed robust reconciliation also demonstrates excellent efficiency when no outliers exist in HTS. Finally, we showcase the practical application of the proposed method in a real-data study on Australian domestic tourism.

Streaming data often exhibit heterogeneity due to heteroscedastic variances or inhomogeneous covariate effects. Online renewable quantile and expectile regression methods provide valuable tools for detecting such heteroscedasticity by combining current data with summary statistics from historical data. However, quantile regression can be computationally demanding because of the non-smooth check function. To address this, we propose a novel online renewable method based on expectile regression, which efficiently updates estimates using both current observations and historical summaries, thereby reducing storage requirements. By exploiting the smoothness of the expectile loss function, our approach achieves superior computational efficiency compared with existing online renewable methods for streaming data with heteroscedastic variances or inhomogeneous covariate effects. We establish the consistency and asymptotic normality of the proposed estimator under mild regularity conditions, demonstrating that it achieves the same statistical efficiency as oracle estimators based on full individual-level data. Numerical experiments and real-data applications demonstrate that our method performs comparably to the oracle estimator while maintaining high computational efficiency and minimal storage costs.

Corner kicks are an important event in soccer because they are often the result of strong attacking play and can be of keen interest to sports fans and bettors. Peng, Hu, and Swartz (2024, Computational Statistics) formulate the mixture feature of corner kick times caused by previous corner kicks, frame the commonly available corner kick data as right-censored event times, and explore patterns of corner kicks. This paper extends their modeling to accommodate the potential correlations between corner kicks by the same teams within the same games. We con- sider a frailty model for event times and apply the Monte Carlo Expec- tation Maximization (MCEM) algorithm to obtain the maximum like- lihood estimates for the model parameters. We compare the proposed model with the model in Peng, Hu, and Swartz (2024) using likelihood ratio tests. The 2019 Chinese Super League (CSL) data are employed throughout the paper for motivation and illustration.

Emerging applications increasingly demand flexible covariate adaptive randomization (CAR) methods that support unequal targeted allocation ratios. While existing procedures can achieve covariate balance, they often suffer from the shift problem. This occurs when the allocation ratios of some additional covariates deviate from the target. We show that this problem is equivalent to a mismatch between the conditional average allocation ratio and the target among units sharing specific covariate values, revealing a failure of existing procedures in the long run. To address it, we derive a new form of allocation function by requiring that balancing covariates ensures the ratio matches the target. Based on this form, we design a class of parameterized allocation functions. When the parameter roughly matches certain characteristics of the covariate distribution, the resulting procedure can balance covariates. Thus, we propose a feasible randomization procedure that updates the parameter based on collected covariate information, rendering the procedure non-Markovian. To accommodate this, we introduce a CAR framework that allows non-Markovian procedure. We then establish its key theoretical properties, including the boundedness of covariate imbalance in probability and the asymptotic distribution of the imbalance for additional covariates. Ultimately, we conclude that the feasible randomization procedure can achieve covariate balance and eliminate the shift.

Causal inference in modern largescale systems faces growing challenges, including highdimensional covariates, multi-valued treatments, massive observational (OBS) data, and limited randomized controlled trial (RCT) samples due to cost constraints. We formalize treatment-induced structural non-overlap and show that, under this regime, commonly used weighted fusion methods provably fail to satisfy randomized identifying restrictions.To address this issue,we propose a constrained joint estimation framework that minimizes observational risk while enforcing causal validity through orthogonal experimental moment conditions. We further show that structural non-overlap creates a feasibility obstruction for moment enforcement in the original covariate space.We also derive a penalized primaldual algorithm that jointly learns representations and predictors, and establish oracle inequalities decomposing error into overlap recovery, moment violation, and statistical terms.Extensive synthetic experiments demonstrate robust performance under varying degrees of nonoverlap. A largescale ridehailing application shows that our method achieves substantial gains over existing baselines, matching the performance of models trained with significantly more RCT data.

A thermodynamic framework for asymptotic inference is developed in which sample size and parameter variance define a state space. Within this description, Shannon information plays the role of entropy, and an integrating factor organizes its variation into a first-law-type balance equation. The framework supports a cyclic inequality analogous to a reversed second law, derived for the estimation of the mean. A non-trivial third-law-type result emerges as a lower bound on entropy set by representation noise. Optimal inference paths, global bounds on information gain, and a natural Carnot-like information efficiency follow from this structure, with efficiency fundamentally limited by a noise floor. Finally, de Bruijn's identity and the I-MMSE relation in the Gaussian-limit case appear as coordinate projections of the same underlying thermodynamic structure. This framework suggests that ensemble physics and inferential physics constitute shadow processes evolving in opposite directions within a unified thermodynamic description.

Learning low-dimensional latent representations is a central topic in statistics and machine learning, and rotation methods have long been used to obtain sparse and interpretable representations. Despite nearly a century of widespread use across many fields, rigorous guarantees for valid inference for the learned representation remain lacking. In this paper, we identify a surprisingly prevalent phenomenon that suggests a reason for this gap: for a broad class of vintage rotations, the resulting estimators exhibit a non-estimable bias. Because this bias is independent of the data, it fundamentally precludes the development of valid inferential procedures, including the construction of confidence intervals and hypothesis testing. To address this challenge, we propose a novel bias-free rotation method within a general representation learning framework based on latent variables. We establish an oracle inference property for the learned sparse representations: the estimators achieve the same asymptotic variance as in the ideal setting where the latent variables are observed. To bridge the gap between theory and computation, we develop an efficient computational framework and prove that its output estimators retain the same oracle property. Our results provide a rigorous inference procedure for the rotated estimators, yielding statistically valid and interpretable representation learning. Code for implementing the proposed method is available at the GitHub repository: https://github.com/chengyu06/Folomin.git.

Time-varying covariates in longitudinal studies frequently evolve through reciprocal feedback, undergo role reversal, and reflect unobserved individual heterogeneity. Standard statistical frameworks often assume fixed covariate roles and exogenous predictors, limiting their utility in systems governed by dynamic behavioral or physiological processes. We develop a hierarchical joint modeling framework that unifies three key features of such systems: (i) bidirectional feedback between a binary and a continuous covariate, (ii) role reversal in which these covariates become jointly modeled outcomes at a prespecified decision phase, and (iii) a shared latent trait influencing both intermediate covariates and a final binary endpoint. The model proceeds in three phases: a feedback-driven longitudinal process, a reversal phase in which the two covariates are jointly modeled conditional on the latent trait, and an outcome model linking a binary, decision-relevant endpoint to observed and latent components. Estimation is carried out using maximum likelihood and Bayesian inference, with Hamiltonian Monte Carlo supporting robust posterior estimation for models with latent structure and mixed outcome types. Simulation studies show that the model yields well calibrated coverage, small bias, and improved predictive performance compared to standard generalized linear mixed models, marginal approaches, and models that ignore feedback or latent traits. In an analysis of nationally representative U.S. panel data, the model captures the co-evolution of physical activity and body mass index and their joint influence, moderated by a latent behavioral resilience factor, on income mobility. The framework offers a flexible, practically implementable tool for analyzing longitudinal decision systems in which feedback, covariate role transition, and unmeasured capacity are central to prediction and intervention.

A conventional Bayesian approach to prediction uses the posterior distribution to integrate out parameters in a density for unobserved data conditional on the observed data and parameters. When the true posterior is intractable, it is replaced by an approximation; here we focus on variational approximations. Recent work has explored methods that learn posteriors optimized for predictive accuracy under a chosen scoring rule, while regularizing toward the prior or conventional posterior. Our work builds on an existing predictive variational inference (PVI) framework that improves prediction, but also diagnoses model deficiencies through implicit model expansion. In models where the sampling density depends on the parameters through a linear predictor, we improve the interpretability of existing PVI methods as a diagnostic tool. This is achieved by adopting PVI posteriors of Gaussian mixture form (GM-PVI) and establishing connections with plug-in prediction for mixture-of-experts models. We make three main contributions. First, we show that GM-PVI prediction is equivalent to plug-in prediction for certain mixture-of-experts models with covariate-independent weights in generalized linear models and hierarchical extensions of them. Second, we extend standard PVI by allowing GM-PVI posteriors to vary with the prediction covariate and in this case an equivalence to plug-in prediction for mixtures of experts with covariate-dependent weights is established. Third, we demonstrate the diagnostic value of this approach across several examples, including generalized linear models, linear mixed models, and latent Gaussian process models, demonstrating how the parameters of the original model must vary across the covariate space to achieve improvements in prediction.

Discrete diffusion models have achieved strong empirical performance in text and other symbolic domains, with masked (absorbing-rate) variants emerging as competitive alternatives to autoregressive models. Among existing samplers, the Euler method remains the standard choice in many applications, and more recently, the First-Hitting Sampler (FHS) has shown considerable promise for masked diffusion models. Despite their practical success, the theoretical understanding of these samplers remains limited. Existing analyses are conducted in Kullback-Leibler (KL) divergence, which often yields loose parameter dependencies and requires strong assumptions on score estimation. Moreover, these guarantees do not cover recently developed high-performance sampler of FHS. In this work, we first develop a direct total-variation (TV) based analysis for the Euler method that overcomes these limitations. Our results relax assumptions on score estimation, improve parameter dependencies, and establish convergence guarantees without requiring any surrogate initialization. Also for this setting, we provide the first convergence lower bound for the Euler sampler, establishing tightness with respect to both the data dimension $d$ and the target accuracy $\varepsilon$. Finally, we analyze the FHS sampler and show that it incurs no sampling error beyond that induced by score estimation, which we show to be tight with a matching lower error bound. Overall, our analysis introduces a direct TV-based error decomposition along the CTMC trajectory and a decoupling-based path-wise analysis for FHS, which may be of independent interest.

In this work, we study scaling limits of shallow Bayesian neural networks (BNNs) via their connection to Gaussian processes (GPs), with an emphasis on statistical modeling, identifiability, and scalable inference. We first establish a general convergence result from BNNs to GPs by relaxing assumptions used in prior formulations, and we compare alternative parameterizations of the limiting GP model. Building on this theory, we propose a new covariance function defined as a convex mixture of components induced by four widely used activation functions, and we characterize key properties including positive definiteness and both strict and practical identifiability under different input designs. For computation, we develop a scalable maximum a posterior (MAP) training and prediction procedure using a Nyström approximation, and we show how the Nyström rank and anchor selection control the cost-accuracy trade-off. Experiments on controlled simulations and real-world tabular datasets demonstrate stable hyperparameter estimates and competitive predictive performance at realistic computational cost.

This work introduces a statistical procedure to infer the interaction graph of neuronal networks modeled by Galves-Löcherbach dynamics. The methodology performs bivariate inference, identifying synaptic links from the spike trains of pairs of neurons without observing the rest of the network. We propose a Macro-Micro Extrapolation algorithm to address data sparsity by inferring interactions in the limit $\Delta \to 0^+$. The core component is a Spike-Triggered Estimator that leverages the local reset property to decouple synaptic jumps from background noise. By employing an adaptive logic that switches between sample averaging and Pyramid Extrapolation, the framework categorizes connections as excitatory, inhibitory, or null. Numerical simulations demonstrate that the classifier identifies synapses without error across varying noise regimes and complex network topologies, even for observation windows broader than those predicted by the current theoretical bounds.

Flow matching has emerged as a simulation-free alternative to diffusion-based generative modeling, producing samples by solving an ODE whose time-dependent velocity field is learned along an interpolation between a simple source distribution (e.g., a standard normal) and a target data distribution. Flow-based methods often exhibit greater training stability and have achieved strong empirical performance in high-dimensional settings where data concentrate near a low-dimensional manifold, such as text-to-image synthesis, video generation, and molecular structure generation. Despite this success, existing theoretical analyses of flow matching assume target distributions with smooth, full-dimensional densities, leaving its effectiveness in manifold-supported settings largely unexplained. To this end, we theoretically analyze flow matching with linear interpolation when the target distribution is supported on a smooth manifold. We establish a non-asymptotic convergence guarantee for the learned velocity field, and then propagate this estimation error through the ODE to obtain statistical consistency of the implicit density estimator induced by the flow-matching objective. The resulting convergence rate is near minimax-optimal, depends only on the intrinsic dimension, and reflects the smoothness of both the manifold and the target distribution. Together, these results provide a principled explanation for how flow matching adapts to intrinsic data geometry and circumvents the curse of dimensionality.

Gradient-boosted decision trees are among the strongest off-the-shelf predictors for tabular regression, but point predictions alone do not quantify uncertainty. Conformal prediction provides distribution-free marginal coverage, yet standard split conformal uses a single global residual quantile and can adapt poorly to heteroscedasticity. We propose LoBoost, a model-native local conformal method that reuses the fitted ensemble's leaf structure to define a multiscale partition of the feature space. Each input is represented by the sequence of leaves it visits along the boosting path, and matching leaf prefixes define nested groups in which residual quantiles are estimated locally. By reusing the predictive structure already learned by the model, LoBoost requires no auxiliary partition or nuisance model, no retraining, and no additional data split beyond standard conformal calibration. Our theory connects the stability of the fitted ensemble and the geometry of its induced cells to local residual-score homogeneity, providing finite-sample coverage-error control and asymptotic pointwise validity as the local calibration size grows and the cells become sufficiently homogeneous. Experiments show competitive interval quality, low post-hoc calibration costs, and stable behavior across local calibration-size settings.

This paper considers a non-standard problem of generating samples from a low-temperature Gibbs distribution with constrained support, when some of the coordinates of the mode lie on the boundary. These coordinates are referred to as the non-regular part of the model. We show that in a “pre-asymptotic” regime in which the limiting Laplace approximation is not yet valid, the low-temperature Gibbs distribution concentrates on a neighborhood of its mode. Within this region, the distribution is a bounded perturbation of a product measure: a strongly log-concave distribution in the regular part and a one-dimensional exponential-type distribution in each coordinate of the non-regular part. Leveraging this structure, we provide a non-asymptotic sampling guarantee by analyzing the spectral gap of Langevin dynamics. Key examples of low-temperature Gibbs distributions include Bayesian posteriors, and we demonstrate our results on three canonical examples: a high-dimensional logistic regression model, a Poisson linear model, and a Gaussian mixture model.

We introduce Multiproposal Elliptical Slice Sampling, a self-tuning multiproposal Markov chain Monte Carlo method for Bayesian inference with Gaussian priors. Our method generalizes the Elliptical Slice Sampling algorithm by 1) allowing multiple candidate proposals to be sampled in parallel at each self-tuning step, and 2) basing the acceptance step on a distance-informed transition matrix that can favor proposals far from the current state. This allows larger moves in state space and faster self-tuning, at essentially no additional wall clock time for expensive likelihoods, and results in improved mixing. We additionally provide theoretical arguments and experimental results suggesting dimension-robust mixing behavior, making the algorithm particularly well suited for Bayesian PDE inverse problems.

Kurtosis-based Independent Component Analysis (ICA) weakens in wide, balanced mixtures. We prove a sharp redundancy law: for a standardized projection with effective width $R_{\mathrm{eff}}$ (participation ratio), the population excess kurtosis obeys $|\kappa(y)|=O(\kappa_{\max}/R_{\mathrm{eff}})$, yielding the order-tight $O(c_b\kappa_{\max}/R)$ under balance (typically $c_b=O(\log R)$). As an impossibility screen, under standard finite-moment conditions for sample kurtosis estimation, surpassing the $O(1/\sqrt{T})$ estimation scale requires $R\lesssim \kappa_{\max}\sqrt{T}$. We also show that purification – selecting $m\!\ll\!R$ sign-consistent sources – restores $R$-independent contrast $\Omega(1/m)$, with a simple data-driven heuristic. Synthetic experiments validate the predicted decay, the $\sqrt{T}$ crossover, and contrast recovery.

We identify a fundamental pathology in the likelihood for time delay inference which challenges standard inference methods. By analysing the likelihood for time delay inference with Gaussian process light curve models, we show that it generically develops a boundary-driven "W"-shape with a global maximum at the true delay and gradual rises towards the edges of the observation window. This arises because time delay estimation is intrinsically extrapolative. In practice, global samplers such as nested sampling are steered towards spurious edge modes unless strict convergence criteria are adopted. We demonstrate this with simulations and show that the effect strengthens with higher data density over a fixed time span. To ensure convergence, we provide concrete guidance, notably increasing the number of live points. Further, we show that methods implicitly favouring small delays, for example optimisers and local MCMC, induce a bias towards larger $H_0$. Our results clarify failure modes and offer practical remedies for robust fully Bayesian time delay inference.

Data privacy is important in the AI era, and differential privacy (DP) is one of the golden solutions. However, DP is typically applicable only if data have a bounded underlying distribution. We address this limitation by leveraging second-moment information from a small amount of public data. We propose Public-moment-guided Truncation (PMT), which transforms private data using the public second-moment matrix and applies a principled truncation whose radius depends only on non-private quantities: data dimension and sample size. This transformation yields a well-conditioned second-moment matrix, enabling its inversion with a significantly strengthened ability to resist the DP noise. Furthermore, we demonstrate the applicability of PMT by using penalized and generalized linear regressions. Specifically, we design new loss functions and algorithms, ensuring that solutions in the transformed space can be mapped back to the original domain. We have established improvements in the models' DP estimation through theoretical error bounds, robustness guarantees, and convergence results, attributing the gains to the conditioning effect of PMT. Experiments on synthetic and real datasets confirm that PMT substantially improves the accuracy and stability of DP models.

Self-attention is usually described as a flexible, content-adaptive way to mix a token with information from its past. We reinterpret causal self-attention transformers, the backbone of modern foundation models, within a probabilistic framework, much as classical PCA is extended to probabilistic PCA. This reformulation reveals a key structural consequence of the underlying change of variables: a barrier constraint emerges on the parameters of self-attention. The resulting geometry exposes a degeneracy boundary where the attention-induced mapping becomes locally ill-conditioned, yielding a stability-margin interpretation analogous to the margin in support vector machines. This, in turn, naturally gives rise to the concept of support tokens. We further show that causal transformers define a consistent stochastic process over infinite token sequences, providing a rigorous probabilistic foundation for sequence modeling. Building on this view, we derive a Bayesian MAP training objective that requires only a minimal modification to standard LLM training: adding a smooth log-barrier penalty to the usual cross-entropy loss. Empirically, the resulting training objective improves robustness to input perturbations and sharpens the margin geometry of the learned representations without sacrificing out-of-sample accuracy.

This paper develops a class of Bayesian non- and semiparametric methods for estimating regression curves and surfaces. The main idea is to model the regression as locally linear, and then place suitable local priors on the local parameters. The method requires the posterior distribution of the local parameters given local data, and this is found via a suitably defined local likelihood function. When the width of the local data window is large the methods reduce to familiar fully parametric Bayesian methods, and when the width is small the estimators are essentially nonparametric. When noninformative reference priors are used the resulting estimators coincide with recently developed well-performing local weighted least squares methods for nonparametric regression. Each local prior distribution needs in general a centre parameter and a variance parameter. Of particular interest are versions of the scheme that are more or less automatic and objective in the sense that they do not require subjective specifications of prior parameters. We therefore develop empirical Bayes methods to obtain the variance parameter and a hierarchical Bayes method to account for uncertainty in the choice of centre parameter. There are several possible versions of the general programme, and a number of its specialisations are discussed. Some of these are shown to be capable of outperforming standard nonparametric regression methods, particularly in situations with several covariates.

The recent article `Satellite conjunction analysis and the false confidence theorem' (Balch, Martin, and Ferson, 2019, Proceedings of the Royal Society, Series A) points to certain difficulties with Bayesian analysis when used for models for satellite conjuntion and ensuing operative decisions. Here we supplement these previous analyses and findings with further insights, uncovering what we perceive of as being the crucial points, explained in a prototype setup where exact analysis is attainable. We also show that a different and frequentist method, involving confidence distributions, is free of the false confidence syndrome.

We study the basic task of mean estimation in the presence of mean-shift contamination. In the mean-shift contamination model, an adversary is allowed to replace a small constant fraction of the clean samples by samples drawn from arbitrarily shifted versions of the base distribution. Prior work characterized the sample complexity of this task for the special cases of the Gaussian and Laplace distributions. Specifically, it was shown that consistent estimation is possible in these cases, a property that is provably impossible in Huber's contamination model. An open question posed in earlier work was to determine the sample complexity of mean estimation in the mean-shift contamination model for general base distributions. In this work, we study and essentially resolve this open question. Specifically, we show that, under mild spectral conditions on the characteristic function of the (potentially multivariate) base distribution, there exists a sample-efficient algorithm that estimates the target mean to any desired accuracy. We complement our upper bound with a qualitatively matching sample complexity lower bound. Our techniques make critical use of Fourier analysis, and in particular introduce the notion of a Fourier witness as an essential ingredient of our upper and lower bounds.

Understanding the geometry of learned distributions is fundamental to improving and interpreting diffusion models, yet systematic tools for exploring their landscape remain limited. Standard latent-space interpolations fail to respect the structure of the learned distribution, often traversing low-density regions. We introduce a framework based on the string method that computes continuous paths between samples by evolving curves under the learned score function. Operating on pretrained models without retraining, our approach interpolates between three regimes: pure generative transport, which yields continuous sample paths; gradient-dominated dynamics, which recover minimum energy paths (MEPs); and finite-temperature string dynamics, which compute principal curves – self-consistent paths that balance energy and entropy. We demonstrate that the choice of regime matters in practice. For image diffusion models, MEPs contain high-likelihood but unrealistic ”cartoon” images, confirming prior observations that likelihood maxima appear unrealistic; principal curves instead yield realistic morphing sequences despite lower likelihood. For protein structure prediction, our method computes transition pathways between metastable conformers directly from models trained on static structures, yielding paths with physically plausible intermediates. Together, these results establish the string method as a principled tool for probing the modal structure of diffusion models – identifying modes, characterizing barriers, and mapping connectivity in complex learned distributions.

Causal identification functionals often require integration over conditional densities of continuous variables, such as those arising in nonparametric identification theory of total and mediated causal effects in DAGs with hidden variables. Estimating these densities and evaluating the resulting integrals can be statistically and computationally demanding. A common workaround is to discretize the continuous variable and replace integrals with finite sums. Although convenient, discretization alters the population-level functional and can induce non-negligible approximation bias, even when identification is correct. Under smoothness conditions, we show that the resulting coarsening error is first order in the bin width and arises at the level of the target functional, distinct from statistical estimation error. We propose a simple debiased coarsened functional that evaluates the outcome regression at within-bin conditional means, eliminating the leading coarsening error term and yielding a second-order approximation error. We derive plug-in and one-step estimators for this debiased coarsened functional. Simulations demonstrate substantial bias reduction and near-nominal confidence interval coverage, even under coarse binning. Our results provide a simple framework for controlling the impact of variable discretization on both parameter approximation and statistical estimation.

When learning interpretable latent structures using model-based approaches, even small deviations from modeling assumptions can lead to inferential results that are not mechanistically meaningful. In this work, we consider latent structures that consist of $K_o$ mechanistic processes, where $K_o$ is unknown. When the model is misspecified, likelihood-based model selection methods can substantially overestimate $K_o$ while more robust nonparametric methods can be overly conservative. Hence, there is a need for approaches that combine the sensitivity of likelihood-based methods with the robustness of nonparametric ones. We formalize this objective in terms of a robust model selection consistency property, which is based on a component-level discrepancy measure that captures the mechanistic structure of the model. We then propose the accumulated cutoff discrepancy criterion (ACDC), which leverages plug-in estimates of component-level discrepancies. To apply ACDC, we develop mechanistically meaningful component-level discrepancies for a general class of latent variable models that includes unsupervised and supervised variants of probabilistic matrix factorization and mixture modeling. We show that ACDC is robustly consistent when applied to unsupervised matrix factorization and mixture models. Numerical results demonstrate that in practice our approach reliably identifies a mechanistically meaningful number of latent processes in numerous illustrative applications, outperforming existing methods.