Approximating discrete probability distributions with causal dependence trees

Chow and Liu considered the problem of approximating discrete joint distributions with dependence tree distributions where the goodness of the approximations were measured in terms of KL distance. They (i) demonstrated that the minimum divergence approximation was the tree with maximum sum of mutual informations, and (ii) specified a low-complexity minimum-weight spanning tree algorithm to find the optimal tree. In this paper, we consider an analogous problem of approximating the joint distribution on discrete random processes with causal, directed, dependence trees, where the approximation is again measured in terms of KL distance. We (i) demonstrate that the minimum divergence approximation is the directed tree with maximum sum of directed informations, and (ii) specify a low-complexity minimum weight directed spanning tree, or arborescence, algorithm to find the optimal tree. We also present an example to demonstrate the algorithm.