Before revision: In this paper, we extend a recently proposed framework for message passing on "loopy" networks to the solution of probabilistic models . We derive a self-consistent set of message passing equations that allow for fast computation of probability distributions in systems that contain short loops, potentially with high density, as well as  expressions for the entropy and partition function of such systems , which are notoriously difficult quantities to compute.
After revision: In this paper, we extend a recently proposed framework for message passing on "loopy" networks to the solution of probabilistic models on networks such as epidemic models, spin models, and Bayesian graphical models, but it suffers from the serious shortcoming that it works poorly in the common case of networks that contain short loops. Here we provide a solution to this long-standing problem, deriving a belief propagation method that allows for fast calculation of probability distributions in systems that contain short loops, potentially with high density, as well as expressions for the entropy and partition function of such systems , which are notoriously difficult quantities to compute.