Compressive Sensing via Variational Bayesian Inference

The sparse signal recovery problem from a set of compressively sensed noisy measurements using sparse Bayesian learning (SBL) modeling and variational Bayesian (VB) inference technique is considered. In the context of SBL, two main approaches are considered here. In the first approach, each component of the sparse signal is modeled via a Gaussian prior with a Gamma/inverse-Gamma hyper prior on its variance/precision. In the second model, each component of the sparse signal is modeled via a Gaussian prior combined with a Bernoulli prior along with a Gamma/inverse-Gamma hyper prior on its variance/precision. In this work, we consider such modeling and derive the update rules for the latent variables and parameters of each modeling in detail. We believe that such rigorous details on these two modeling and inferences provide sufficient intuition for better understanding the inference using variational Bayes, which can also serve as basic models when incorporating any further structures on the sparse/compressible signal.