I generated a list of sequence observations (artificial) using an HMM model. Then I tried to infer the HMM model parameter by training it with the observation list using the Viterbi and EM algorithms. At each step I inferred the hidden sequences and updated the model until the sum of likelihoods of all the training examples were stable and no longer increased. Using this method, the final model is far away from the original one. I used a left-to-right HMM model and I initialized my start model randomly, but in such way that it would be near to the original model. Does anyone have any suggestions as to why the HMM training using the Viterbi and EM algorithms doesn't converge?
The way you describe it is a little vague so forgive me if I misunderstand some things. Firstly, you say that the sum of likelihoods is stable and doesn't increase. This sounds like 'convergence' to me, despite the fact that it obviously didn't converge to the right answer.
A better method would be to use the Baum-Welch algorithm. A short description is given on wikipedia: http://en.wikipedia.org/wiki/Baum%E2%80%93Welch_algorithm
It uses EM+Forward Backward, not EM+Viterbi. This distinction is important because you need to be able to localize the precise probabilities at a given point in time. What the Viterbi algorithm gives (the best total sequence) is not useful for training.
Note however that there is no exact algorithm to infer model parameters for an arbitrary HMM. You are far better off finding some structure in your data and leveraging that.
Alireza has some good points. Viterbi training is only justified over Baum-Welch under some very specific circumstances. It can be a way to "focus in" when multiple interpretations are otherwise allowed. When assessing the quality of a fit, you should also consider whether your HMM has some symmetry structure, so that alternative assignments of the state IDs are almost equivalent.
In the end, naturally, the quality of a fit should be judged based on its intended use. The fact that you are using Viterbi might indicate that your end goal is also to reconstruct the single most likely state sequence for new observations. Is it really the full sequence you need, or to know e.g. simply how much time is spent in some specific state?
Thank you Alireza and Carl for your answers and I'm sorry if my question was a little bit ambiguous. In the first step, I tried Viterbi algorithm to see what it can give as result, but it seems that it's not good for HMM training. I introduced an new idea in the way of updating the model parameters, i don't use just a simple counting (for example for transitions or emissions ), I exploit the probabilities that Viterbi attaches to each state in the best total sequence), but it still gives a bad result. The next step, as you mentioned, is to use the Baum-Welch algorithm and see its results.
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