Error bounds for some approximate posterior measures in Bayesian inference

In certain applications involving the solution of a Bayesian inverse problem, it may not be possible or desirable to evaluate the full posterior, e.g. due to the high computational cost. This problem motivates the use of approximate posteriors that arise from approximating the negative log-likelihood or forward model. We review some error bounds for random and deterministic approximate posteriors that arise when the approximate negative log-likelihoods and approximate forward models are random.