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Automatic Tempered Posterior Distributions for Bayesian Inversion Problems

2021; Multidisciplinary Digital Publishing Institute; Volume: 9; Issue: 7 Linguagem: Inglês

10.3390/math9070784

2227-7390

Luca Martino, Fernando Llorente, Ernesto Curbelo, J. López‐Santiago, Joaquı́n Mı́guez,

Gaussian Processes and Bayesian Inference

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 are carried out using distinct (but interacting) methods. 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 with 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 current estimate of the noise power. A complete Bayesian study over the model parameters and the scale parameter can also be performed. Numerical experiments show the benefits of the proposed approach.