Automatic tempered posterior distributions for Bayesian inversion problems
This work addresses Bayesian inversion problems for researchers in computational statistics or inverse modeling, but it appears incremental as it builds on existing methods by combining Bayesian and maximum likelihood approaches.
The authors tackled Bayesian inversion problems by splitting inference of model parameters and noise power, using an adaptive importance sampling scheme that alternates sampling and optimization steps, resulting in a sequence of automatically tempered posterior densities with demonstrated benefits in numerical experiments.
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.