Computing The Kullback-Leibler Divergence Between Probabilistic Automata Using Rational Kernels

Kullback-Leibler divergence is a natural distance measure between two probabilistic finite-state automata. Computing this distance is difficult, since it requires a summation over a countably infinite number of strings. Nederhof and Satta (2004) recently provided a solution in the course of solving the more general problem of finding the cross-entropy between a probabilistic context-free grammar and an unambiguous probabilistic automaton. We propose a novel solution for two unambiguous probabilistic automata, by showing that Kullback-Leibler divergence can be defined as a rational kernel (Cortes et al., 2004) over the expectation semiring (Eisner, 2002). Using this definition, the computation is performed using the general algorithm for rational kernels, yielding an elegant and efficient solution.