Numerical posterior distribution error control and expected Bayes Factors in the bayesian Uncertainty Quantification of Inverse Problems

Abstract

In the bayesian analysis of Inverse Problems most relevant cases the forward maps (FM, or regressor function) are defined in terms of a system of (O, P)DE's with intractable solutions. These necessarily involve a numerical method to find approximate versions of such solutions and lead to a numerical/approximate posterior distribution. Recently several results have been published on the regularity conditions required on such numerical methods to ensure converge of the numerical to the theoretical posterior. However, more practical guidelines are needed to ensure a suitable working numerical posterior. ]Capistran2016] prove for ODE's that the Bayes Factor of the approximate vs the theoretical model tends to 1 in the same order as the numerical method order. In this work we generalize the latter paper in that we consider 1) also PDE's, 2) correlated observations, 3) practical guidelines in a multidimensional setting and 4) explore the use of expected Bayes Factors. This permits us to obtain bounds on the absolute global errors to be tolerated by the FM numerical solver, which we illustrate with some examples. Since the Bayes Factor is kept above 0.95 we expect that the resulting numerical posterior is basically indistinguishable from the theoretical posterior, even though we are using an approximate numerical FM. The method is illustrated with some examples using synthetic data.