Hidden Markov Model
You should bear in mind clearly the three questions people usually ask for Hidden Markov Model:
1. what is the probability of an observed sequence?
2. what is the most likely series of states given a specific observed observation?
3. Given a set of observations, what are the values of the state transition probabilities and the output emission probabilities?
ref:
http://cs229.stanford.edu/section/cs229-hmm.pdf
http://mlg.eng.cam.ac.uk/zoubin/papers/ijprai.pdf