Adaptive posterior distributions for covariance matrix learning in Bayesian inversion problems for multioutput signals

In this work, we propose an adaptive importance sampling (AIS) scheme for multivariate Bayesian inversion problems, which is based in two main ideas: the inference procedure is divided in two parts and the variables of interest are split in two blocks. We assume that the observations are generated from a complex multivariate non-linear function perturbed by correlated Gaussian noise. We estimate both the unknown parameters of the multivariate non-linear model and the covariance matrix of the noise. In the first part of the proposed inference scheme, a novel AIS technique called adaptive target AIS (ATAIS) is designed, which alternates iteratively between an IS technique over the parameters of the non-linear model and a frequentist approach for the covariance matrix of the noise. In the second part of the proposed inference scheme, a prior density over the covariance matrix is considered and the cloud of samples obtained by ATAIS are recycled and re-weighted for obtaining a complete Bayesian study over the model parameters and covariance matrix. Two numerical examples are presented that show the benefits of the proposed approach.