Bayesian compressive sensing using Monte Carlo methods

The problem of reconstructing a signal from compressively sensed measurements is solved in this work from a Bayesian perspective. The proposed reconstruction solution differs from previous Bayesian methods in that it numerically evaluates the posterior of the sparse solution. This allows the method to utilize any kind of information on the signal without the need to evaluate the posterior in closed form. Specifically, the method uses multi-stage sampling together with a greedy subroutine to efficiently draw information directly from the likelihood and any prior distribution on the signal, including a sparsity prior. The approach is shown to accurately represent the Bayesian belief on the sparse solution based on noisy compressively sensed signals.