Analytic Formulas for Renyi Entropy of Hidden Markov Models

Abstract

Determining entropy rates of stochastic processes is a fundamental and difficult problem, with closed-form solutions known only for specific cases. This paper pushes the state-of-the-art by solving the problem for Hidden Markov Models (HMMs) and Renyi entropies. While the problem for Markov chains reduces to studying the growth of a matrix product, computations for HMMs involve \emph{products of random matrices}. As a result, this case is much harder and no explicit formulas have been known so far. We show how to circumvent this issue for Renyi entropy of integer orders, reducing the problem again to a \emph{single matrix products} where the matrix is formed from transition and emission probabilities by means of tensor product. To obtain results in the asymptotic setting, we use a novel technique for determining the growth of non-negative matrix powers. The classical approach is the Frobenius-Perron theory, but it requires positivity assumptions; we instead work directly with the spectral formula. As a consequence, our results do not suffer from limitations such as irreducibility and aperiodicity. This improves our understanding of the entropy rate even for standard (unhidden) Markov chains. A recently published side-channel attack against RSA was proven effective using our result, specialized to order 2.