Model selection is a longstanding problem in statistics wherein the goal is to infer which subset of predictors best explains the observed data. Once this predictor subset has been inferred, regression can be accomplished in a straightforward manner. Sparse reconstruction, a key component of compressive sensing, faces exactly the same problem, i.e., inferring which subset of (typically linear) model coefficients are non-zero. Once the subset has been inferred, estimation of the non-zero coefficient values is easily accomplished. Bayesian approaches to model selection have been popular within the statistics community for some time and, more recently, have been popularized for sparse reconstruction. We review some of the recent work in this area, describe connections to matching pursuit algorithms, and show some promising numerical results. In addition, we discuss ongoing work to analyze the performance of these Bayesian algorithms.