I have a sequence of two possible observations ($A$, $B$) and want to train an HMM with $h$ states, namely $\lambda_h$, to predict the probability of the next observation using the Baum-Welch algorithm. Output probability distributions are Bernoulli with parameter $p_i$ ($i$ is the state number).
My problem is mainly with the initialization. If $\lambda_h$ is initialized to a totally random HMM, results of learning are poor. On the other hand, the K-Means initialization algorithm assigns zero initial probability ($\pi_i$) to some states if $h > 2$, preventing BW from improvements. I use a 4-state HMM ($\lambda^0_4$) to generate my test sequences. Hence trying $h=4$ should be totally possible!
Any ideas?
Parameters of $\lambda^0_4$:
Transition probabilities: $A_{i,i}$ = 0.4, $A_{i,j\neq i}$=0.2
Observation probability distributions: $p_0$=1.0, $p_1$=0.7, $p_2$=0.3, $p_3$=0.0
Initial state probabilities: $\pi_i$ = 0.25
Best Answer
You might want to try the rule of succession. Just add $1/n$ to all your $\pi_i$ (where $n$ is the number of training examples).
This is, of course a heuristic. But K-Means initialization is one as well.