# Score-Based Martingale Posteriors for Deep Neural Networks

arXiv: https://arxiv.org/abs/2606.15725 · HTML: https://mlsanity.com/paper/2606.15725/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 19 references that are in the map.
- First posted on arXiv 14 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [Preconditioned Stochastic Gradient Descent](https://mlsanity.com/paper/1512.04202/) (2015)
- [Martingale posterior distributions](https://mlsanity.com/paper/2103.15671/) (2021)
- [Martingale Posterior Neural Processes](https://mlsanity.com/paper/2304.09431/) (2023)
- [TabMGP: Martingale Posterior with TabPFN](https://mlsanity.com/paper/2510.25154/) (2025)
- [Bayesian Deep Learning with Multilevel Trace-class Neural Networks](https://mlsanity.com/paper/2203.12961/) (2022)
- [Unbiased Estimation using Underdamped Langevin Dynamics](https://mlsanity.com/paper/2206.07202/) (2022)
- [Martingale Posteriors from Score Functions](https://mlsanity.com/paper/2501.01890/) (2025)
- [New Trends in the Stability of Sinkhorn Semigroups](https://mlsanity.com/paper/2601.12633/) (2026)

## Abstract (arXiv)

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.