# Probability and log probability in hidden Markov models

I have a set of Observation Symbol Sequences which I have to test against a set of Trained HMM classifiers. I seem to understand the advantages of using Log Probability over regular probabilities.

In the testing phase of a HMM classifier, I don't seem to get the motivation behind multiplying probabilities or adding log probabilities in determining the class of the observed observation test sequence.

Why do we have to multiply or add probabilities? Can't we just determine probability or log probability of a Single Observation Symbol Sequence using Forward algorithm?

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A Markov Model has probabilities for each individual transition (the transfer function). In the case of a Hidden Markov Model (HMM) there is also a probability function mapping the hidden state(s) to observations.

These probabilities have to be combined to produce the sequence probability. Therefore they are multiplied. (and for Log probabilities, this operator then becomes an addition)

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I am determining the probability of entire observation sequence (one symbol sequence at a time). I was wondering If I have to combine the probabilities of multiple sequences. I'm using jahmm library. –  garak Feb 10 '12 at 15:17
Sorry I don't know enough about your specific application (and jahmm is new to me) - I thought you were talking about adding the probabilities within a sequence. –  winwaed Feb 10 '12 at 15:19

Yes, the probability of an observation sequence can be computed using the forward algorithm.

Note however that the forward algorithm is an iterative algorithm where a bunch of summations and multiplications are carried out in each iteration. So the answer to your second question is yes as well.

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