Kullback-Leibler Distance between Probabilistic Context-Free Grammars and Probabilistic Finite Automata

We consider the problem of computing the Kullback-Leibler distance, also called the relative entropy, between a probabilistic context-free grammar and a probabilistic finite automaton. We show that there is a closed-form (analytical) solution for one part of the Kullback-Leibler distance, viz. the cross-entropy. We discuss several applications of the result to the problem of distributional approximation of probabilistic context-free grammars by means of probabilistic finite automata.