  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.
