Asymptotics of the entropy rate for a hidden Markov process

We calculate the Shannon entropy rate of a binary hidden Markov process (HMP), of given transition rate and noise ε (emission), as a series expansion in ε. The first two orders are calculated exactly. We then evaluate, for finite histories, simple upper-bounds of Cover and Thomas. Surprisingly, we find that for a fixed order k and history of n steps, the bounds become independent of n for large enough n. This observation is the basis of a conjecture, that the upper-bound obtained for n≥(k+3)/2 gives the exact entropy rate for any desired order k of ε.