, 
, and 
, we compute the
probability of the input sequence 
.
By this algorithm, we can evaluate the probability that a given HMM
model produced a given observation sequence. Given an HMM model M ,
consisting of , , and , we compute the
probability of the input sequence .
First, we define 
as the probability of generating the
partial sequence 
, ending up in state j at time t
. 
is initialized to 1.0 in the initial state, and 0.0
in all other states. If we have already computed 
for
all i in the previous time frame t-1, then can be
computed recursively in terms of the incremental probability of
entering state j from each i while generating the output
symbol 
(see Figure 2.5):
図 2.5: Forward pass recursion in HMM
If F is the final state, then we see that 
is the
probability that the HMM generated the complete output sequence
.
