Finding the probability of an observation I
Given the observation sequence O=(o1, …, oT) and a model ?= (A, B, ?), we wish to know how to efficiently compute P(O| ?). This process is called decoding.
For any state sequence X=(X1, …, XT+1), we find: P(O|?)=? X1…XT+1 ?X1 ?t=1T aXtXt+1 bXtXt+1ot
This is simply the sum of the probability of the observation occurring according to each possible state sequence.
Direct evaluation of this expression, however, is extremely inefficient.