A computational method for bounding the probability of reconstruction on trees

Abstract: For a tree Markov random field non-reconstruction is said to hold if as the depth of the tree goes to infinity the information that a typical configuration at the leaves gives about the value at the root goes to zero. The distribution of the measure at the root conditioned on a typical boundary can be computed using a distributional recurrence. However the exact computation is not feasible because the support of the distribution grows exponentially with the depth. In this work, we introduce a notion of a survey of a distribution over probability vectors which is a succinct representation of the true distribution. We show that a survey of the distribution of the measure at the root can be constructed by an efficient recursive algorithm. The key properties of surveys are that the size does not grow with the depth, they can be constructed recursively, and they still provide a good bound for the distance between the true conditional distribution and the unconditional distribution at the root. This approach applies to a large class of Markov random field models including randomly generated ones. As an application we show bounds on the reconstruction threshold for the Potts model on small-degree trees.