How to evaluate the goodness of Fit of parameters obtained from EM algorithm

Question

I have a set of observations $\mathcal{Y} = {Y_1, \ldots, Y_T}$. I am running EM algorithm to fit the observations to the following Hidden Markov Model

$$A = [a_{ij}]_{N \times N}, a_{ij} = P(X_{k+1} = j \mid X_k = i), i,j = 1, \ldots,N$$ $$B = [b_{ij}]_{N \times N}, b_{ij} = P(Y_k = j \mid X_k = i), i, j = 1, \ldots, N$$

More specifically, I run the EM algorithm to estimate the probability transition matrix (of the underlying markov chain) and the observation probability matrix.

I want to know how can I estimate the goodness of fit of the obtained estimates $\hat{A}, \hat{B}$.

Answer 1

For one possible answer, see this paper http://people.stat.sfu.ca/~raltman/myinfo/biom2004.pdf