Diffusion models [1, 2, 3] have emerged as a powerful class of generative models, achieving state-of-the-art performance across a wide range of applications, including imaging [2] and scientific-data synthesis [4]. From a statistical perspective, they can be viewed as flexible nonparametric estimators of a (conditional) distribution via score estimation and reverse-time stochastic differential equations (SDEs) [5, 6]. Despite this expressive power, standard diffusion models are typically causality-agnostic: they learn a joint law without encoding the directional asymmetries required for causal interpretation. As a consequence, they do not, on their own, provide principled answers to interventional queries or support broader causal analyses, which are central to structural causal models (SCMs) [7]. When a causal ordering (or a directed acyclic graph) is available, it is natural to construct generative procedures that sample variables sequentially according to the causal factorisation. Such iterative, ordering-respecting approaches have been proposed using a variety of generative models, including generative adversarial networks [8], variational autoencoders [9], normalising flows [10], and diffusion-based constructions such as DDIM [11]. However, a rigorous statistical understandingof the advantages of exploitingsuch causalstructureand the inferential use of the resulting generator remain less developed.

Bayesian Optimal Experimental Design (BOED) provides a rigorous framework for decision-making tasks in which data acquisition is often the critical bottleneck, especially in resource-constrained settings. Traditionally, BOED typically selects designs by maximizing expected information gain (EIG), commonly defined through the Kullback-Leibler (KL) divergence. However, classical evaluation of EIG often involves challenging nested expectations, and even advanced variational methods leave the underlying log-density-ratio objective unchanged. As a result, support mismatch, tail underestimation, and rare-event sensitivity remain intrinsic concerns for KL-based BOED. To address these fundamental bottlenecks, we introduce an IPM-based BOED framework that replaces density-based divergences with integral probability metrics (IPMs), including the Wasserstein distance, Maximum Mean Discrepancy, and Energy Distance, resulting in a highly flexible plug-and-play BOED framework. We establish theoretical guarantees showing that IPM-based utilities provide stronger geometry-aware stability under surrogate-model error and prior misspecification than classical EIG-based utilities. We also validate the proposed framework empirically, demonstrating that IPM-based designs yield highly concentrated credible sets. Furthermore, by extending the same sample-based BOED template in a plug-and-play manner to geometry-aware discrepancies beyond the IPM class, illustrated by a neural optimal transport estimator, we achieve accurate optimal designs in high-dimensional settings where conventional nested Monte Carlo estimators and advanced variational methods fail.

We explore the top-K rank aggregation problem in which one aims to recover a consistent ordering that focuses on top-K ranked items based on partially revealed preference information. We examine an M-wise comparison model that builds on the Plackett-Luce (PL) model where for each sample, M items are ranked according to their perceived utilities modeled as noisy observations of their underlying true utilities. As our result, we characterize the minimax optimality on the sample size for top-K ranking. The optimal sample size turns out to be inversely proportional to M. We devise an algorithm that effectively converts M-wise samples into pairwise ones and employs a spectral method using the refined data. In demonstrating its optimality, we develop a novel technique for deriving tight $\ell_\infty$ estimation error bounds, which is key to accurately analyzing the performance of top-K ranking algorithms, but has been challenging. Recent work relied on an additional maximum-likelihood estimation (MLE) stage merged with a spectral method to attain good estimates in $\ell_\infty$ error to achieve the limit for the pairwise model. In contrast, although it is valid in slightly restricted regimes, our result demonstrates a spectral method alone to be sufficient for the general M-wise model. We run numerical experiments using synthetic data and confirm that the optimal sample size decreases at the rate of 1/M. Moreover, running our algorithm on real-world data, we find that its applicability extends to settings that may not fit the PL model.

Transformer-based scientific foundation models are increasingly deployed in high-stakes settings, but current architectures give deterministic outputs and provide limited support for calibrated predictive uncertainty. We propose Stochastic Attention, a lightweight inference-time modification that randomizes attention by replacing softmax weights with normalized multinomial samples controlled by a single concentration parameter, and produces predictive ensembles without retraining. To set this parameter, we introduce a calibration objective that matches the stochastic attention output with the target, yielding an efficient univariate post-hoc tuning problem. We evaluate this mechanism on two scientific foundation models for weather and timeseries forecasting along with an additional regression task. Across benchmarks against uncertainty-aware baselines, we find that Stochastic Attention achieves the strongest native calibration and the sharpest prediction intervals at comparable coverage, while requiring only minutes of post-hoc tuning versus days of retraining for competitive baselines.

We introduce a principled probabilistic framework for reward-guided decoding in large language models, addressing the limitations of standard decoding methods that optimize token-level likelihood rather than sequence-level quality. Our method defines a reward-augmented target distribution over complete sequences by combining model transition probabilities with prefix-dependent reward potentials. Importantly, the approach is training-free: it leaves model weights unchanged and instead modifies the inference distribution via reward potentials, with all gains arising purely from inference-time sampling. To sample from this distribution, we develop Sequential Monte Carlo algorithms, including a computationally efficient prefix-only variant and a lookahead variant whose intermediate targets match the exact marginals of the full sequence distribution. The framework also integrates resample-move updates with Metropolis-Hastings rejuvenation and supports block-wise generation, subsuming common decoding strategies such as temperature sampling and power-tempered objectives. Empirical results across three 7B models show significant gains. On code generation (HumanEval), our method improves base performance by up to 54.9% and surpasses the strongest sampling baselines by 9.1%-15.3%. On mathematical reasoning (MATH500), it achieves gains of up to 8.8%. Notably, it reaches 87.8% on HumanEval and 78.4% on MATH500 with Qwen2.5-7B, consistently outperforming the reinforcement learning method GRPO.

Selection bias arises when the probability that an observation enters a dataset depends on variables related to the quantities of interest, leading to systematic distortions in estimation and uncertainty quantification. For example, in epidemiological or survey settings, individuals with certain outcomes may be more likely to be included, resulting in biased prevalence estimates with potentially substantial downstream impact. Classical corrections, such as inverse-probability weighting or explicit likelihood-based models of the selection process, rely on tractable likelihoods, which limits their applicability in complex stochastic models with latent dynamics or high-dimensional structure. Simulation-based inference enables Bayesian analysis without tractable likelihoods but typically assumes missingness at random and thus fails when selection depends on unobserved outcomes or covariates. Here, we develop a bias-aware simulation-based inference framework that explicitly incorporates selection into neural posterior estimation. By embedding the selection mechanism directly into the generative simulator, the approach enables amortized Bayesian inference without requiring tractable likelihoods. This recasting of selection bias as part of the simulation process allows us to both obtain debiased estimates and explicitly test for the presence of bias. The framework integrates diagnostics to detect discrepancies between simulated and observed data and to assess posterior calibration. The method recovers well-calibrated posterior distributions across three statistical applications with diverse selection mechanisms, including settings in which likelihood-based approaches yield biased estimates. These results recast the correction of selection bias as a simulation problem and establish simulation-based inference as a practical and testable strategy for parameter estimation under selection bias.

Classical mixture models (MMs) are widely used tractable proposals for approximate inference settings such as variational inference (VI) and importance sampling (IS). Recently, mixture models with negative coefficients, called subtractive mixture models (SMMs), have been proposed as a potentially more expressive alternative. However, how to effectively use SMMs for VI and IS is still an open question as they do not provide latent variable semantics and therefore cannot use sampling schemes for classical MMs. In this work, we study how to circumvent this issue by designing several expectation estimators for IS and learning schemes for VI with SMMs, and we empirically evaluate them for distribution approximation. Finally, we discuss the additional challenges in estimation stability and learning efficiency that they carry and propose ways to overcome them. Code is available at: https://github.com/april-tools/delta-vi.

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

In variety testing, multi-environment trials (MET) are essential for evaluating the genotypic performance of crop plants. A persistent challenge in the statistical analysis of MET data is the estimation of variance components, which are often still inaccurately estimated or shrunk to exactly zero when using residual (restricted) maximum likelihood (REML) approaches. At the same time, institutions conducting MET typically possess extensive historical data that can, in principle, be leveraged to improve variance component estimation. However, these data are rarely incorporated sufficiently. The purpose of this paper is to address this gap by proposing a Bayesian framework that systematically integrates historical information to stabilize variance component estimation and better quantify uncertainty. Our Bayesian linear mixed model (BLMM) reformulation uses priors and Markov chain Monte Carlo (MCMC) methods to maintain the variance components as positive, yielding more realistic distributional estimates. Furthermore, our model incorporates historical prior information by managing MET data in successive historical data windows. Variance component prior and posterior distributions are shown to be conjugate and belong to the inverse gamma and inverse Wishart families. While Bayesian methodology is increasingly being used for analyzing MET data, to the best of our knowledge, this study comprises one of the first serious attempts to objectively inform priors in the context of MET data. This refers to the proposed Bayesian updating approach. To demonstrate the framework, we consider an application where posterior variance component samples are plugged into an A-optimality experimental design criterion to determine the average optimal allocations of trials to agro-ecological zones in a sub-divided target population of environments (TPE).

In online clustering problems, there is often a large amount of uncertainty over possible cluster assignments that cannot be resolved until more data are observed. This difficulty is compounded when clusters follow complex distributions, as is the case with text data. Sequential Monte Carlo (SMC) methods give a natural way of representing and updating this uncertainty over time, but have prohibitive memory requirements for large-scale problems. We propose a novel SMC algorithm that decomposes clustering problems into approximately independent subproblems, allowing a more compact representation of the algorithm state. Our approach is motivated by the knowledge base construction problem, and we show that our method is able to accurately and efficiently solve clustering problems in this setting and others where traditional SMC struggles.