# Discrete gradients for computational Bayesian inference

## TL;DR

This paper explores the use of discrete gradient methods to efficiently solve stiff differential equations arising in continuous-time Bayesian inference, specifically in ensemble Kalman-Bucy filters and Fokker-Planck discretizations.

## Contribution

It introduces discrete gradient techniques for Bayesian inference, providing a comparative analysis with existing numerical methods for stiff differential equations.

## Key findings

- Discrete gradient methods improve numerical stability.
- They outperform semi-implicit methods in certain scenarios.
- Enhanced efficiency in Bayesian filtering applications.

## Abstract

In this paper, we exploit the gradient flow structure of continuous-time formulations of Bayesian inference in terms of their numerical time-stepping. We focus on two particular examples, namely, the continuous-time ensemble Kalman-Bucy filter and a particle discretisation of the Fokker-Planck equation associated to Brownian dynamics. Both formulations can lead to stiff differential equations which require special numerical methods for their efficient numerical implementation. We compare discrete gradient methods to alternative semi-implicit and other iterative implementations of the underlying Bayesian inference problems.