A Monte Carlo approach to quantifying model error in Bayesian parameter estimation

Quantifying the discrepancy between two distributions is considered, using the concept of ϕ-divergence. The motivation is a Bayesian inference scenario where one is interested in comparing different posterior distributions. Strongly consistent estimators for the ϕ-divergence between two posterior distributions are developed. The proposed estimators alleviate known computational difficulties with estimating normalizing constants. This approach can be used to study the impact that using an approximate likelihood has on the resulting posterior distribution and also to compare the effectiveness of different model approximations. The methodology is applied to two first-order emulator models and an oceanographic application where evaluation of the likelihood function involves the solution to a partial differential equation.