Since the turn of the century, approximate Bayesian inference has steadily evolved as new computational techniques have been incorporated to handle increasingly complex and large-scale predictive problems. The recent success of deep neural networks and foundation models has now given rise to a new paradigm in statistical modeling, in which Bayesian inference can be amortized through large-scale learned predictors. In amortized inference, substantial computation is invested upfront to train a neural network that can subsequently produce approximate posterior or predictions at negligible marginal cost across a wide range of tasks. At deployment, amortized inference offers substantial computational savings compared with traditional Bayesian procedures, which generally require repeated likelihood evaluations or Monte Carlo simulations for predictions for each new dataset. Despite the growing popularity of amortized inference, its statistical interpretation and its role within Bayesian inference remain poorly understood. This paper presents statistical perspectives on the working principles of several major neural architectures, including feedforward networks, Deep Sets, and Transformers, and examines how these architectures naturally support amortized Bayesian inference. We discuss how these models perform structured approximation and probabilistic reasoning in ways that yield controlled generalization error across a wide range of deployment scenarios, and how these properties can be harnessed for Bayesian computation. Through simulation studies, we evaluate the accuracy, robustness, and uncertainty quantification of amortized inference under varying signal-to-noise ratios and distributional shifts, highlighting both its strengths and its limitations.

Multiple binary responses arise in many modern data-analytic problems. Although fitting separate logistic regressions for each response is computationally attractive, it ignores shared structure and can be statistically inefficient, especially in high-dimensional and class-imbalanced regimes. Low-rank models offer a natural way to encode latent dependence across tasks, but existing methods for binary data are largely likelihood-based and focus on pointwise classification rather than ranking performance. In this work, we propose a unified framework for learning with multiple binary responses that directly targets discrimination by minimizing a surrogate loss for the area under the ROC curve (AUC). The method aggregates pairwise AUC surrogate losses across responses while imposing a low-rank constraint on the coefficient matrix to exploit shared structure. We develop a scalable projected gradient descent algorithm based on truncated singular value decomposition. Exploiting the fact that the pairwise loss depends only on differences of linear predictors, we simplify computation and analysis. We establish non-asymptotic convergence guarantees, showing that under suitable regularity conditions, leading to linear convergence up to the minimax-optimal statistical precision. Extensive simulation studies demonstrate that the proposed method is robust in challenging settings such as label switching and data contamination and consistently outperforms likelihood-based approaches.

We propose a novel method for sampling from unnormalized Boltzmann densities based on a probability-flow ordinary differential equation (ODE) derived from linear stochastic interpolants. The key innovation of our approach is the use of a sequence of Langevin samplers to enable efficient simulation of the flow. Specifically, these Langevin samplers are employed (i) to generate samples from the interpolant distribution at intermediate times and (ii) to construct, starting from these intermediate times, a robust estimator of the velocity field governing the flow ODE. For both applications of the Langevin diffusions, we establish convergence guarantees. Extensive numerical experiments demonstrate the efficiency of the proposed method on challenging multimodal distributions across a range of dimensions, as well as its effectiveness in Bayesian inference tasks.

This work introduces a novel technique, named structural dimension reduction, to collapse a Bayesian network onto a minimum and localized one while ensuring that probabilistic inferences between the original and reduced networks remain consistent. To this end, we propose a new combinatorial structure in directed acyclic graphs called the directed convex hull, which has turned out to be equivalent to their minimum localized Bayesian networks. An efficient polynomial-time algorithm is devised to identify them by determining the unique directed convex hulls containing the variables of interest from the original networks. Experiments demonstrate that the proposed technique has high dimension reduction capability in real networks, and the efficiency of probabilistic inference based on directed convex hulls can be significantly improved compared with traditional methods such as variable elimination and belief propagation algorithms. The code of this study is open at \href{https://github.com/Balance-H/Algorithms}{https://github.com/Balance-H/Algorithms} and the proofs of the results in the main body are postponed to the appendix.

The $ρ$-posterior framework provides universal Bayesian estimation with explicit contamination rates and optimal convergence guarantees, but has remained computationally difficult due to an optimization over reference distributions that precludes intractable posterior computation. We develop a PAC-Bayesian framework that recovers these theoretical guarantees through temperature-dependent Gibbs posteriors, deriving finite-sample oracle inequalities with explicit rates and introducing tractable variational approximations that inherit the robustness properties of exact $ρ$-posteriors. Numerical experiments demonstrate that this approach achieves theoretical contamination rates while remaining computationally feasible, providing the first practical implementation of $ρ$-posterior inference with rigorous finite-sample guarantees.

The classical optimal transport (OT) problem seeks the map that moves mass from a source to a target measure while minimizing a prescribed cost function. The objective can be formalized in either Monge's [12] or Kantronich's formulation [10], a convex relaxation of the former that considers transport plans instead of deterministic maps. These foundational formulations have wide-ranging applications, including to economics [7] and machine learning [14]. In many practical scenarios, the source measure is known or readily in-ferrable from empirical data but the target measure is not explicitly specified. Instead, it is only constrained by practical requirements or expert knowledge. For example, when applying Monge's formulation to transportation problems, the placement of the mass in the target region may be constrained to lie entirely beyond a certain boundary or within a particular region, rather than by the specification of a precise location for each fraction of the total mass. Similarly, in economic applications, supply and demand may be subject to constraints such as maximal amounts available or minimal amounts required, rather than dictated through precise marginal distributions. 1

Score-based diffusion models have become a powerful framework for generative modeling, with score estimation as a central statistical bottleneck. Existing guarantees for score estimation largely focus on light-tailed targets or rely on restrictive assumptions such as compact support, which are often violated by heavy-tailed data in practice. In this work, we study conventional (Gaussian) score-based diffusion models when the target distribution is heavy-tailed and belongs to a Sobolev class with smoothness parameter $β>0$. We consider both exponential and polynomial tail decay, indexed by a tail parameter $γ$. Using kernel density estimation, we derive sharp minimax rates for score estimation, revealing a qualitative dichotomy: under exponential tails, the rate matches the light-tailed case up to polylogarithmic factors, whereas under polynomial tails the rate depends explicitly on $γ$. We further provide sampling guarantees for the associated continuous reverse dynamics. In total variation, the generated distribution converges at the minimax optimal rate $n^{-β/(2β+d)}$ under exponential tails (up to logarithmic factors), and at a $γ$-dependent rate under polynomial tails. Whether the latter sampling rate is minimax optimal remains an open question. These results characterize the statistical limits of score estimation and the resulting sampling accuracy for heavy-tailed targets, extending diffusion theory beyond the light-tailed setting.

Inference-time alignment for diffusion models aims to adapt a pre-trained diffusion model toward a target distribution without retraining the base score network, thereby preserving the generative capacity of the base model while enforcing desired properties at the inference time. A central mechanism for achieving such alignment is guidance, which modifies the sampling dynamics through an additional drift term. In this work, we introduce Doob's matching, a novel framework for guidance estimation grounded in Doob's $h$-transform. Our approach formulates guidance as the gradient of logarithm of an underlying Doob's $h$-function and employs gradient-penalized regression to simultaneously estimate both the $h$-function and its gradient, resulting in a consistent estimator of the guidance. Theoretically, we establish non-asymptotic convergence rates for the estimated guidance. Moreover, we analyze the resulting controllable diffusion processes and prove non-asymptotic convergence guarantees for the generated distributions in the 2-Wasserstein distance.

Meta-learning methods for regression like Neural (Diffusion) Processes achieve impressive results, but with these models it can be difficult to incorporate expert prior knowledge and information contained in metadata. Large Language Models (LLMs) are trained on giant corpora including varied real-world regression datasets alongside their descriptions and metadata, leading to impressive performance on a range of downstream tasks. Recent work has extended this to regression tasks and is able to leverage such prior knowledge and metadata, achieving surprisingly good performance, but this still rarely matches dedicated meta-learning methods. Here we introduce a general method for sampling from a product-of-experts of a diffusion or flow matching model and an `expert' with binned probability density; we apply this to combine neural diffusion processes with LLM token probabilities for regression (which may incorporate textual knowledge), exceeding the empirical performance of either alone.

Neural networks have shown state-of-the-art performances in various classification and regression tasks. Rectified linear units (ReLU) are often used as activation functions for the hidden layers in a neural network model. In this article, we establish the connection between the Poisson hyperplane processes (PHP) and two-layer ReLU neural networks. We show that the PHP with a Gaussian prior is an alternative probabilistic representation to a two-layer ReLU neural network. In addition, we show that a two-layer neural network constructed by PHP is scalable to large-scale problems via the decomposition propositions. Finally, we propose an annealed sequential Monte Carlo algorithm for Bayesian inference. Our numerical experiments demonstrate that our proposed method outperforms the classic two-layer ReLU neural network. The implementation of our proposed model is available at https://github.com/ShufeiGe/Pois_Relu.git.