Fast Bayesian Model Selection in Imaging Inverse Problems Using Residuals

This paper presents a fast heuristic for comparing Bayesian models to solve inverse problems related to signal processing. We focus on problems that are convex w.r.t. the unknown signal and where no ground truth is available. The proposed heuristic is very computationally efficient and does not require the estimation of the model evidence. Instead, the model evidence is used indirectly to set the regularisation parameters that define each competing model by maximum marginal likelihood estimation, followed by a simple likelihood-based or residual-based comparison of the models based on their empirical Bayesian maximum-a-posteriori solutions. The proposed methodology is illustrated with a total-variation image deblurring experiment, where it performs remarkably well.