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Milestones in sparse signal reconstruction and compressive sensing can be understood in a probabilistic Bayesian context, fusing underdetermined measurements with knowledge about low-level signal properties in the posterior distribution, which is maximized for point estimation. We review recent progress to advance beyond this setting. If the posterior is used as a distribution to be integrated over instead of merely an optimization criterion, sparse estimators with better properties may be obtained, and applications beyond point reconstruction from fixed data can be served. We describe novel variational relaxations of Bayesian integration, characterized as well as posterior maximization, which can be solved robustly for very large models by algorithms unifying convex reconstruction and Bayesian graphical model technology. They excel in difficult real-world imaging problems where posterior maximization performance is often unsatisfactory.