Accuracy of posterior approximations via χ2 and harmonic divergences

Unlike the more familiar precision estimates based on Central Limit type theorems for Monte Carlo based π̂, 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 π̂, 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.