  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.
