The Entropy Rate of the Hidden Markov Process

A stochastic process which is a noisy observation of a Markov process through a memoryless channel is called a hidden Markov process (HMP). Finding the entropy rate of HMP is motivated by applications in both stochastic signal processing and information theory. The first expression for the entropy rate of HMP is found in 1957 [1]. This expression is defined through a measure described by an integral equation which is hard to extract from the equation in any explicit way. Besides a known bound on the entropy rate for the general case, simple expressions for the entropy rate has been recently obtained for special cases where the parameters of hidden Markov source approaches zero [2],[3]. In this paper the entropy rate is defined by a measure as the stationary distribution of a Markov process with specific initial distribution and conditional probabilities. This allows exact computation of state distribution for any time index, and eventually the measure. Although the state space of the associated Markov process is a continuous set, we have shown that its distribution is a probability mass function, which leads to a simple algorithm to generate a sequence of numbers that converges monotonically from above to the entropy rate. A Hidden Markov Process {Zn}∞n=0 is defined by a quadruple [S, P,Z, T ], where S, P are the state set and transition probability matrix of a Markov process and Z, T is the observation set and the observation probability Matrix |S| × |Z|, respectively. By defining the random variable qn(Zn−1) over the |Z|-dimensional probability simplex (denoted by ∇Z) with components qn[k] = Pr(Zn = k|Zn−1), we show that the entropy rate is Ĥ(Z) = limn→∞E[h(qn)], where h(.) is the entropy function over ∇Z . Since {qn}∞n=0 is not a Markov process, the measure for evaluating this expectation is not computable. However if we consider the information-state process {πn}∞n=0, which is related to {qn}∞n=0 by the transformation qn = q(πn) = πn × T , the entropy rate can be formulated as