Bayesian inference in linear models with a random Gaussian matrix : Algorithms and complexity

We consider the Bayesian inference of a random Gaussian vector in a linear model with a random Gaussian matrix. We review two approaches to finding the MAP estimator for this model. We propose improved versions of these approaches with reduced complexity. Next we analyze their complexity and convergence properties. Then we derive the MAP estimator in the setting in which the variance of the noise is unknown. Simulation results presented compare the performance in terms of estimation error of the approaches.