We present a new local approximation algorithm for computing Maximum a Posteriori (MAP) and log-partition function for arbitrary exponential family distribution represented by a finite-valued pair-wise Markov random field (MRF), say G. Our algorithm is based on decomposition of G into {em appropriately} chosen small components; then computing estimates locally in each of these components and then producing a {em good} global solution. We show that if the underlying graph G either excludes some finite-sized graph as its minor (e.g. Planar graph) or has low doubling dimension (e.g. any graph with {em geometry}), then our algorithm will produce solution for both questions within {em arbitrary accuracy}. We present a message-passing implementation of our algorithm for MAP computation using self-avoiding walk of graph. In order to evaluate the computational cost of this implementation, we derive novel tight bounds on the size of self-avoiding walk tree for arbitrary graph.
As a consequence of our algorithmic result, we show that the normalized log-partition function (also known as free-energy) for a class of {em regular} MRFs will converge to a limit, that is computable to an arbitrary accuracy.