Title: The Bayesian update: variational formulations and gradient flows,
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.