Error bounds for some approximate posterior measures in Bayesian inference

TL;DR

Addresses the computational cost of Bayesian inverse problems by establishing error bounds for approximate posterior measures arising from random misfits or random forward models. The analysis, based on the Hellinger distance $d_{\mathrm{H}}$, links posterior error to the misfit or forward-model approximation error under exponential tail and integrability conditions. For random misfits, the main result bounds $\mathbb{E}_{\nu_N}[d_{\mathrm{H}}(\mu,\mu_N)^2]^{1/2}$ by a norm of $|\Phi-\Phi_N|$, with a parallel bound for the marginal posterior $\mu^{\mathrm{M}}_N$. For random forward models, nonasymptotic bounds relate $d_{\mathrm{H}}(\mu,\mu^{\mathrm{M}}_N)$ and $d_{\mathrm{H}}(\mu,\mu_N)$ to $\mathbb{E}_{\nu_N}[\|G_N-G\|^{p}]$ for suitable $p$, under exponential integrability. The results support the use of randomized numerical methods in Bayesian inference and guide construction of accurate approximate posteriors.

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.

Theorems & Definitions (5)

• theorem 1: Error bound for random approximate posterior
• theorem 2: Error bound for marginal approximate posterior
• corollary 1: Joint conditions for error bounds on both approximate posteriors
• remark 1
• theorem 3: Error bounds for approximate posteriors