Posterior distribution existence and error control in Banach spaces in the Bayesian approach to UQ in inverse problems

TL;DR

This paper extends Bayesian inverse problem analysis to Banach spaces, providing bounds on numerical errors to ensure accurate posterior distributions and establishing existence and convergence results in infinite-dimensional settings.

Contribution

It generalizes posterior existence and error control results to Banach spaces, including finite-dimensional approximations, in Bayesian inverse problems.

Findings

Bound on numerical error for posterior accuracy

Existence of infinite-dimensional posterior distribution

Convergence rates for discretized Bayesian inverse problems

Abstract

We generalize the results of \cite{Capistran2016} on expected Bayes factors (BF) to control the numerical error in the posterior distribution to an infinite dimensional setting when considering Banach functional spaces and now in a prior setting. The main result is a bound on the absolute global error to be tolerated by the Forward Map numerical solver, to keep the BF of the numerical vs. the theoretical model near to 1, now in this more general setting, possibly including a truncated, finite dimensional approximate prior measure. In so doing we found a far more general setting to define and prove existence of the infinite dimensional posterior distribution than that depicted in, for example, \cite{Stuart2010}. Discretization consistency and rates of convergence are also investigated in this general setting for the Bayesian inverse problem.