On Monte Carlo computation of posterior expectations with uncertainty

In this paper, we study computation of the range of posterior expectations that arise from robust Bayesian statistics. We compute supremum and infimum of the posterior expectations, when allowing uncertainty for the choice of the likelihood function, or uncertainty for the choice of the prior distribution. In the standard approach of sensitivity analysis, posterior statistics is computed a multiple number of times for each choice of the uncertainty scenarios, which might involve heavy computation due to running Monte Carlo sampling many times. Our paper proposes a more efficient computational method that only requires one Monte Carlo sample for all possible choices of the uncertainty scenarios. The proposed computational method involves three steps (with the mnemonic PSI): (Prior step.) Introduce an auxiliary hyperprior distribution on a parameter λ that indexes the uncertainty. (Sampling step.) For any parameter of interest h, we derive a sample of (h, λ) from the joint posterior distribution given the observed data, using a Monte Carlo method. ¹ (Inference step.) Based on this posterior sample of (h, λ), we estimate the range of posterior expectations {inf_λ E(h|λ, data), sup_λ E(h|λ, data)} (and similarly the range for any posterior quantile).