Error Control of the Numerical Posterior with Bayes Factors in Bayesian Uncertainty Quantification

In this paper, we address the numerical posterior error control problem for the Bayesian approach to inverse problems or recently known as Bayesian Uncertainty Quantification (UQ). We generalize the results of Capistra??n et al. (2016) to (a priori) expected Bayes factors (BF) and in a more general, infinite dimensional setting. In this inverse problem, the unavoidable numerical approximation of the Forward Map (FM, i.e., the regressor function), arising from the numerical solution of a system of differential equations, demands error estimates of the corresponding approximate numerical posterior distribution. Our approach is to make such comparisons in the setting of Bayesian model selection and BFs. The main result of this paper is a bound on the absolute global error tolerated by the numerical solver of the FM in order to keep the BF of the numerical versus the theoretical posterior near one. For two examples, we provide a detailed analysis of the computation and implementation of the introduced bound. Furthermore, we show that the resulting numerical posterior turns out to be nearly identical from the theoretical posterior, given the control of the BF near one.