Abstract

We derive an expression for marginals of arbitrary order of a tree-structured distribution in terms pairwise correlations. The expression involves an expansion into terms of varying order, each one being described by a certain combinatorial optimization. The key property of this expression is that it does not depend explicitly on the structure of the tree. Superficially similar representations make use of tree-recursions to express probabilities in a way that depends on the tree structure. An implication of our result is that two tree-distributions with similar correlations are close in all low- order marginals, a statement of significance to learning tree models in order to subsequently make predictions.