An equivalence between high dimensional Bayes optimal inference and M-estimation

Due to the computational difficulty of performing MMSE (minimum mean squared error) inference, maximum a posteriori (MAP) is often used as a surrogate. However, the accuracy of MAP is suboptimal for high dimensional inference, where the number of model parameters is of the same order as the number of samples. In this work we demonstrate how MMSE performance is asymptotically achievable via optimization with an appropriately selected convex penalty and regularization function which are a smoothed version of the widely applied MAP algorithm. Our findings provide a new derivation and interpretation for recent optimal M-estimators discovered by El Karoui, et. al. PNAS 2013 as well as extending to non-additive noise models. We demonstrate the performance of these optimal M-estimators with numerical simulations. Overall, at the heart of our work is the revelation of a remarkable equivalence between two seemingly very different computational problems: namely that of high dimensional Bayesian integration, and high dimensional convex optimization. In essence we show that the former computationally difficult integral may be computed by solving the latter, simpler optimization problem.