A computational method for bounding the probability of reconstruction on trees

For a tree Markov random field, nonreconstruction 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 as follows: 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.