Q-Hermite polynomials chaos approximation of likelihood function based on q-Gaussian prior in Bayesian inversion
For engineers solving inverse problems, this work provides a theoretical foundation for using q-Gaussian priors and an accelerated sampling method, though the practical impact is incremental.
The paper introduces q-Gaussian priors for Bayesian inversion and proposes an acceleration algorithm using spectral likelihood approximation, proving convergence of the posterior distribution in Kullback-Leibler divergence, total variation, and Hellinger metric, with numerical examples.
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.