Q-Hermite polynomials chaos approximation of likelihood function based on q-Gaussian prior in Bayesian inversion

TL;DR

This paper introduces a q-Gaussian prior and a spectral likelihood approximation method for Bayesian inversion, analyzing convergence of the posterior distribution and demonstrating effectiveness through numerical examples.

Contribution

It proposes a novel q-Gaussian prior and an acceleration algorithm based on spectral likelihood approximation for Bayesian inversion.

Findings

Convergence of the posterior in Kullback-Leibler divergence is established.

Convergence in total variation and Hellinger metric is demonstrated.

Numerical examples validate the proposed method.

Abstract

In real applications, the construction of prior and acceleration of sampling for posterior are usually two key points of Bayesian inversion algorithm for engineers. In this paper, q-analogy of Gaussian distribution, q-Gaussian distribution, is introduced as the prior of inverse problems. And an acceleration algorithm based on spectral likelihood approximation is discussed. We mainly focus on the convergence of the posterior distribution in the sense of Kullback-Leibler divergence when approximated likelihood function and truncated prior distribution are used. Moreover, the convergence in the sense of total variation and Hellinger metric is obtained. In the end two numerical examples are displayed.