Score-Based Martingale Posteriors for Deep Neural Networks

A new expository paper submitted on 14 June 2026 examines score-based martingale posteriors as a computationally faster alternative to conventional Bayesian methods for quantifying uncertainty in deep neural networks [1]. The paper, posted to the arXiv preprint repository, investigates the efficacy of score-based martingale posteriors (SMP) in modern, large-scale machine learning contexts [1]. The SMP approach, building on work by Cui & Walker in 2025 and Fong et al. in 2023, relies on a stochastic gradient ascent-type recursion on the parameter space of stochastic models [1]. Under simple mathematical assumptions, the recursion constructs a martingale sequence whose limiting random variable can be simulated very quickly, in contrast to Monte Carlo-based methods such as Markov chain Monte Carlo (MCMC) [2]. Traditional Bayesian uncertainty quantification models unseen data conditional on observed data using prior distributions, a framework that often requires computationally intensive MCMC algorithms [3]. The martingale posterior approach departs from this paradigm. It constructs convergent sequences of parameters by imputing data from the predictive model with current parameter values, relying on score functions and martingales to establish convergence [3]. Because the method does not require MCMC, it can be implemented in parallel and only needs to sample the model, which is typically straightforward [3]. The authors explore SMP specifically for inferring the parameters of deep neural networks and, where feasible, compare results to state-of-the-art Monte Carlo methods aimed at inferring conventional Bayesian posteriors [4]. The investigation includes experiments on a toy binary classification problem and the MNIST dataset, benchmarking SMP against Bayesian MCMC using the No-U-Turn Sampler and point estimates such as the NUTS mean and maximum a posteriori [5]. The two experiments reveal distinct regimes where SMP can succeed or struggle, and the paper highlights practical guidelines for its application to deep neural networks [5]. The work appears on arXiv, an open-access repository of electronic preprints that, as of November 2024, receives about 24,000 articles per month and hosts over two million papers across mathematics, physics, computer science, and related fields [9]. The paper is accessible through the arXiv abstract page, which also features arXivLabs, a framework allowing community collaborators to develop experimental tools such as bibliographic explorers and code finders directly on the site [7][8].