Foundations of Data Science

a ij
transition probability from state i to state j
b j (O t+1 ) probability of O t+1 given that the HMM is in state j at time t + 1
α t (i)
β t+1 (j)
δ(i, j)
s t (i)
p(O)
probability of seeing O 0 O 1 · · · O t and ending in state i at time t
probability of seeing the tail of the sequence O t+2 O t+3 · · · O T given state j
at time t + 1
probability of going from state i to state j at time t given the sequence
of outputs O
probability of being in state i at time t given the sequence of outputs O
probability of output sequence O
The probability of being in state i at time t is given by
n∑
s t (i) = δ t (i, j).
j=1
Note that δ t (i, j) is the probability of being in state i at time t given O 0 O 1 O 2 · · · O t but
it is not the probability of being in state i at time t given O since it does not take into
account the remainder of the sequence O. Summing s t (i) over all time periods gives the
expected number of times state i is visited and the sum of δ t (i, j) over all time periods
gives the expected number of times edge i to j is traversed.
Given estimates of the HMM parameters a i,j and b j (O k ), we can calculate by the above
formulas estimates for
1. ∑ T −1
i=1 s t(i), the expected number of times state i is visited and departed from
2. ∑ T −1
i=1 δ t(i, j), the expected number of transitions from state i to state j
Using these estimates we can obtain new estimates of the HMM parameters
a ij =
expected number of transitions from state i to state j
expected number of transitions out of state i
b j (O k ) = expected number of times in state j observing symbol O k
expected number of times in state j
307
=
∑ T −1
t=1 δ t(i, j)
∑ T −1
t=1 s t(i)
=
T∑
−1
t=1
subject to
O t=O k
s t (j)
∑ T −1
t=1 s t(j)