# The Bayesian update: variational formulations and gradient flows

## TL;DR

This paper explores the variational and gradient flow perspectives of Bayesian updates, introducing new tools for analyzing convergence and proposing novel MCMC proposal strategies based on these insights.

## Contribution

It formalizes the connection between Bayesian posteriors, variational functionals, and gradient flows, and introduces a criterion for metric choice in Riemannian MCMC methods.

## Key findings

- Convergence rates are bounded by geodesic convexity of the functionals.
- Gradient flows lead to nonlinear diffusions with the posterior as invariant distribution.
- Proposed a criterion for metric selection in Riemannian MCMC.

## Abstract

The Bayesian update can be viewed as a variational problem by characterizing the posterior as the minimizer of a functional. The variational viewpoint is far from new and is at the heart of popular methods for posterior approximation. However, some of its consequences seem largely unexplored. We focus on the following one: defining the posterior as the minimizer of a functional gives a natural path towards the posterior by moving in the direction of steepest descent of the functional. This idea is made precise through the theory of gradient flows, allowing to bring new tools to the study of Bayesian models and algorithms. Since the posterior may be characterized as the minimizer of different functionals, several variational formulations may be considered. We study three of them and their three associated gradient flows. We show that, in all cases, the rate of convergence of the flows to the posterior can be bounded by the geodesic convexity of the functional to be minimized. Each gradient flow naturally suggests a nonlinear diffusion with the posterior as invariant distribution. These diffusions may be discretized to build proposals for Markov chain Monte Carlo (MCMC) algorithms. By construction, the diffusions are guaranteed to satisfy a certain optimality condition, and rates of convergence are given by the convexity of the functionals. We use this observation to propose a criterion for the choice of metric in Riemannian MCMC methods.