In this paper we investigate the efficacy of the score-based martingale posteriors (SMP) (Cui & Walker, 2025; Fong et al., 2023) in the context of modern and large-scale machine learning problems and its potential for meaningful uncertainty quantification. SMPs work with a stochastic gradient ascent-type recursion on the parameter space of stochastic models and construct a martingale on the parameter space. Under simple mathematical assumptions, the recursion can be built so that the parameters form a martingale sequence which possesses a limiting, in time, random variable, the latter of which can be simulated very quickly, in contrast to Monte Carlo-based methods such as Markov chain Monte Carlo. In this expository paper we explore the SMP for inferring the parameters of deep neural networks (DNNs) and, where feasible, compare our results to the state-of-the-art Monte Carlo methods aimed at inferring conventional Bayesian posteriors.

We propose a flexible framework for modeling the predictive distributions of nonlinear, possibly multivariate time series. Our approach expresses a general predictive distribution in an appropriate generative representation that is based on a folklore result from measure theoretic probability. This representation provides a direct simulation-based approximation to the predictive distribution, enabling straightforward computation of forecasts for the conditional mean and variance, fan charts, value at risk, expected shortfall, joint tail risks, and other quantities of interest. We estimate this generative representation using a version of conditional generative adversarial networks and provide a formal statistical analysis of estimation under weak temporal dependence. Specifically, estimation is expressed as a particular minimax problem and we establish consistency of its approximate solutions in Hausdorff distance. The empirical relevance of the approach is illustrated using applications to equity returns, realized variance, and realized covariances. The proposed method is also computationally manageable, with estimation in our applications taking approximately one minute on a standard laptop.

We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.

In the era of Explainable Artificial Intelligence, there is a renewed focus on single trees for their ease of interpretation. This paper introduces Simultaneous Latent Budget Trees, a probabilistic machine learning framework for classification trees in the presence of a stratification factor such as a temporal, spatial, or demographic variable, acting as a control variable or potential confounder. Standard tree growth procedures are not designed to optimize a conditional split rule. A model-based split rule is proposed in which child nodes are interpreted as latent components of a simultaneous mixture model, such as the Simultaneous Latent Budget Model and its constrained versions, fitted to the parent node. Mixing parameters drive the observations, differently for each group, to the child nodes whereas latent budgets parameters update the response classes profile of each level of the control variable. Parameters are estimated by least squares considering a neural network perspective of the model. An informative tree structure can be interactively visualized with interpretation aids on the node and the paths, including visual pruning and decision tree selection procedure. Suitable measures are proposed to handle an unbalanced response class distribution. The proposed methodology is applied to investigate gender-related differences in disease progression of Amyotrophic Lateral Sclerosis. The SLBT library with the various tree-based algorithms is available in the linked GitHub repository.

This paper explores policy gradient algorithms for training stochastic policies to sample from structured discrete probability distributions under the Generative Flow Network (GFlowNet) framework. Building on extensive theoretical connections between GFlowNets and entropy-regularized reinforcement learning, we derive equivalents of standard policy gradient algorithms for training GFlowNets, as well as experimentally explore their various methodological aspects, including baseline training and advantage estimation. Most importantly, our work is the first to derive and successfully apply proximal policy optimization to GFlowNets, showing its improved convergence speed and data efficiency compared to standard GFlowNet training objectives on benchmarks ranging from synthetic energies to molecular graph generation.

A fundamental challenge in cognitive neuroscience is understanding how cognition emerges from the interplay between structural connectivity (SC) and functional connectivity (FC). Current machine learning approaches typically seek to establish direct mappings from SC to FC associated with specific cognitive states. However, these methods often treat SC and FC as distinct endpoints, failing to capture the coupling relationship throughout the progressive transformation between them. To address this limitation, we propose BrainFlow, a reversible generative model designed to parametrize flows between the distribution of SC and the mixed distribution of FCs from different cognitive tasks.

Recent research shows that in-context learning (ICL) can be effective even when demonstrations have missing or incorrect labels. To shed light on this capability, we examine a canonical setting where the demonstrations are drawn according to a binary Gaussian mixture model (GMM) and a certain fraction of the demonstrations have missing labels.

Conventional representation learning methods learn a universal representation that primarily captures dominant semantics, which may not always align with customized downstream tasks. For instance, in animal habitat analysis, researchers prioritize scene-related features, whereas universal embeddings emphasize categorical semantics, leading to suboptimal results. As a solution, existing approaches resort to supervised fine-tuning, which however incurs high computational and annotation costs. In this paper, we propose Conditional Representation Learning (CRL), aiming to extract representations tailored to arbitrary user-specified criteria. Specifically, we reveal that the semantics of a space are determined by its basis, thereby enabling a set of descriptive words to approximate the basis for a customized feature space. Building upon this insight, given a user-specified criterion, CRL first employs a large language model (LLM) to generate descriptive texts to construct the semantic basis, then projects the image representation into this conditional feature space leveraging a vision-language model (VLM). The conditional representation better captures semantics for the specific criterion, which could be utilized for multiple customized tasks. Extensive experiments on classification and retrieval tasks demonstrate the superiority and generality of the proposed CRL.

Distinguishing cause and effect from bivariate observational data is a foundational problem in many disciplines, but challenging without additional assumptions. Additive noise models (ANMs) are widely used to enable sample-efficient bivariate causal discovery. However, conventional ANM-based methods fail when unobserved mediators corrupt the causal relationship between variables. This paper makes three key contributions: first, we rigorously characterize why standard ANM approaches break down in the presence of unmeasured mediators. Second, we demonstrate that prior solutions for hidden mediation are brittle in finite sample settings, limiting their practical utility. To address these gaps, we propose Bivariate Denoising Diffusion (BiDD) for causal discovery, a method designed to handle latent noise introduced by unmeasured mediators. Unlike prior methods that infer directionality through mean squared error loss comparisons, our approach introduces a novel independence test statistic: during the noising and denoising processes for each variable, we condition on the other variable as input and evaluate the independence of the predicted noise relative to this input. We prove asymptotic consistency of BiDD under the ANM, and conjecture that it performs well under hidden mediation. Experiments on synthetic and real-world data demonstrate consistent performance, outperforming existing methods in mediator-corrupted settings while maintaining strong performance in mediator-free settings.