Entropy Rate for Hidden Markov Chains with rare transitions

We consider Hidden Markov Chains obtained by passing a Markov Chain with rare transitions through a noisy memoryless channel. We obtain asymptotic estimates for the entropy of the resulting Hidden Markov Chain as the transition rate is reduced to zero. Let (Xn) be a Markov chain with finite state space S and transition matrix P (p) and let (Yn) be the Hidden Markov chain observed by passing (Xn) through a homogeneous noisy memoryless channel (i.e. Y takes values in a set T , and there exists a matrix Q such that P(Yn = j|Xn = i,Xn−1 −∞ , X∞ n+1, Y n−1 −∞ , Y∞ n+1) = Qij). We make the additional assumption on the channel that the rows of Q are distinct. In this case we call the channel statistically distinguishing. Finally we assume that P (p) is of the form I + pA where A is a matrix with negative entries on the diagonal, non-negative entries in the off-diagonal terms and zero row sums. We further assume that for small positive p, the Markov chain with transition matrix P (p) is irreducible. Notice that for Markov chains of this form, the invariant distribution (πi)i∈S does not depend on p. In this case, we say that for small positive values of p, the Markov chain is in a rare transition regime. We will adopt the convention that H is used to denote the entropy of a finite partition, whereas h is used to denote the entropy of a process (the entropy rate in information theory terminology). Given an irreducible Markov chain with transition matrix P , we let h(P ) be the entropy of the Markov chain (i.e. h(P ) = − ∑ i,j πiPij logPij where πi is the (unique) invariant distribution of the Markov chain and where as usual we adopt the convention that 0 log 0 = 0). We also let Hchan(i) be the entropy of the output of the channel when the input symbol is i (i.e. Hchan(i) = − ∑ j∈T Qij logQij ). Let h(Y ) denote the entropy of Y (i.e. h(Y ) = − limN→∞ 1 N ∑ w∈TN P(Y N 1 = w) log P(Y N 1 = w)) Theorem 1. Consider the Hidden Markov Chain (Yn) obtained by observing a Markov chain with irreducible transition matrix P (p) = I+Ap through a statistically distinguishing channel with transition matrix Q. Then there exists a constant C > 0 such that for all small p > 0,