Entropy rate calculations of algebraic measures

Abstract

Abstract-Let K = {0, 1, ..., q − 1}. We use a special class of translation invariant measures on K Z called algebraic measures to study the entropy rate of a hidden Markov processes. Under some irreducibility assumptions of the Markov transition matrix we derive exact formulas for the entropy rate of a general q state hidden Markov process derived from a Markov source corrupted by a specific noise model. We obtain upper bounds on the error when using an approximation to the formulas and numerically compute the entropy rates of two and three state hidden Markov models.

An efficient algorithm for the entropy rate of a hidden Markov model with unambiguous symbols

“…In [11] we showed that the support of the Blackwell measure for the hidden Markov model described by the noise model in this section is countable.…”

“…In [11] we considered a specific noise model which we call a hidden Markov model with at least one unambiguous received symbol. If the symbol 0 is transmitted then it is always received as 0 at the other end.…”

“…Proposition 2.1 ( [11]). For the HMP with one or more unambiguous received symbol the support of the measure φ is given by…”

“…Next we state the assumptions and statement of the main theorem from [11] for the entropy rate of the hidden Markov process under consideration. Let p = min ij e ij and P = max ij e ij .…”

“…Algebraic measures were introduced by Fannes, Nachtergaele and Werner in the context of quantum spin systems as classical analogues of finitely correlated states and were shown to be in one to one correspondence with functions of Markov processes or hidden Markov processes. In [11] we used the approach of algebraic measures to compute the entropy rate of a hidden Markov model with at least one unambiguous symbol and showed that an approximation to the formula converges exponentially fast to the entropy rate. Our paper is organized as follows; in section 2 we give background about the entropy rate problem, introduce the noise model and review the results of [11], in section 3 we show an efficient algorithm to compute the entropy rate and present numerical estimates of the entropy rate using a sequence of observed symbols and in section 4 we use the results to derive bounds on the capacity of the Gilbert channel.…”