In order to illustrate some comments made to my previous post, I suggest
the following homework problem:
We are dealing with a Hidden Markov model with n internal states, which produces as output a sequence of Hs and Ts. We know that it is ergodic (*), we are told that it is biased such that either p(H) > 2p(T) or p(T) > 2p(H) (if it just runs long enough) and we observe a sequence HTHHT... of N ouptputs.
How big does N have to be as a function of n to determine with some confidence if we are
dealing with one or the other case?
If we do not know n (**), how large would N have to be?
And how does an uninformed prior look like in this case? Will it have to be an improper prior as long as we do not know n or should we somehow weight with the (inverse of) the complexity of the HMMs?
(*) Every internal state can eventually be reached and every internal state
will eventually be left, but the transition probabilities might be small.
(**) This makes it somewhat similar to the case of the previous blog post, but of course, in the previous post we do not know anything about the underlying model, other than that there is a strong bias.