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 of doing so. This problem motivates the use of approximate posteriors that arise from approximating the data misfit or forward model. We review some error bounds for random and deterministic approximate posteriors that arise when the approximate data misfits and approximate forward models are random.