Structured sparsity has recently emerged in statistics, machine learning and signal processing as a promising paradigm for learning in high-dimensional settings. A number of methods have been proposed for learning under the assumption of structured sparsity, including group LASSO and graph LASSO. All of these methods rely on prior knowledge on how to weight (equivalently, how to penalize) individual subsets of variables during the subset selection process. However, these weights on groups of variables are in general unknown. Inferring group weights from data is a key open problem in research on structured sparsity. In this paper, we propose a Bayesian approach to the problem of group weight learning. We model the group weights as hyperparameters of heavy-tailed priors on groups of variables and derive an approximate inference scheme to infer these hyperparameters. We empirically show that we are able to recover the model hyperparameters when the data are generated from the model, and moreover, we demonstrate the utility of learning group weights in synthetic and real denoising problems.