  We propose a data-driven, coarse-graining formulation in the context of
equilibrium statistical mechanics. In contrast to existing techniques which are
based on a fine-to-coarse map, we adopt the opposite strategy by prescribing a
probabilistic coarse-to-fine map. This corresponds to a directed probabilistic
model where the coarse variables play the role of latent generators of the fine
scale (all-atom) data. From an information-theoretic perspective, the framework
proposed provides an improvement upon the relative entropy method and is
capable of quantifying the uncertainty due to the information loss that
unavoidably takes place during the CG process. Furthermore, it can be readily
extended to a fully Bayesian model where various sources of uncertainties are
reflected in the posterior of the model parameters. The latter can be used to
produce not only point estimates of fine-scale reconstructions or macroscopic
observables, but more importantly, predictive posterior distributions on these
quantities. Predictive posterior distributions reflect the confidence of the
model as a function of the amount of data and the level of coarse-graining. The
issues of model complexity and model selection are seamlessly addressed by
employing a hierarchical prior that favors the discovery of sparse solutions,
revealing the most prominent features in the coarse-grained model. A flexible
and parallelizable Monte Carlo - Expectation-Maximization (MC-EM) scheme is
proposed for carrying out inference and learning tasks. A comparative
assessment of the proposed methodology is presented for a lattice spin system
and the SPC/E water model.
