Automatic tempered posterior distributions for Bayesian inversion problems

TL;DR

This paper introduces an adaptive importance sampling method for Bayesian inversion that separates variable inference from noise estimation, using a tempering approach driven by noise power estimates, demonstrated through numerical experiments.

Contribution

It presents a novel iterative scheme combining sampling and optimization, with automatic tempering based on noise power estimation for improved Bayesian inversion.

Findings

Enhanced accuracy in Bayesian inversion results

Automatic selection of tempering parameters improves convergence

Numerical experiments validate the approach's effectiveness

Abstract

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise is split. More specifically, we consider a Bayesian analysis for the variables of interest (i.e., the parameters of the model to invert), whereas we employ a maximum likelihood approach for the estimation of the noise power. The whole technique is implemented by means of an iterative procedure, alternating sampling and optimization steps. Moreover, the noise power is also used as a tempered parameter for the posterior distribution of the the variables of interest. Therefore, a sequence of tempered posterior densities is generated, where the tempered parameter is automatically selected according to the actual estimation of the noise power. A complete Bayesian study over the model parameters and the scale parameter can be also performed. Numerical experiments show the benefits of the proposed approach.