Error bounds for some approximate posterior measures in Bayesian inference

TL;DR

This paper reviews error bounds for approximate posterior measures in Bayesian inverse problems, especially when using simplified models or data misfits to reduce computational costs.

Contribution

It provides a comprehensive review of error bounds for both random and deterministic approximate posteriors in Bayesian inference.

Findings

Error bounds depend on the approximation quality of models and data misfits.

Both random and deterministic approximations are analyzed.

The review guides the selection of approximate methods in Bayesian inference.

Abstract

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.