Accuracy of posterior approximations via χ2 and harmonic divergences

Abstract We propose to use χ2 and Harmonic divergences as global measures of accuracy of an approximation π to a posterior density of interest π. We prove some inequalities which relate these measures to the precision of the corresponding approximations for posterior expectations. In practice these divergences ought to be approximated somehow, and here we propose importance sampling type estimates based on a sample from π . Unlike the more familiar precision estimates based on Central Limit type theorems for Monte Carlo based π , our proposal (i) can be applied to approximations obtained from virtually every method available; (ii) requires to compute only one measure of accuracy which can then be reused to assess precision of the approximations for many posterior expectations and (iii) since its rationale is external to the method used to obtain π , it avoids the danger of circular reasoning present for instance in Markov chain Monte Carlo algorithms, whereby both the validity of the approximation and of its estimated precision depend on convergence of the simulated chain, which in practice may be difficult to assess.