Description:Common Formulas in Hidden Markov Models (HMMs) with outlined proofs.

1. Nidhal Selmi,
Tanishka Singh
Hidden Markov Models
common probabilities
nselmi@asu.edu
Spring 2019
HMM Diagram:
Latent Q: { q1 qt
si
qt+1
sj
qt+2 qT }
o1 ot ot+1 ot+2 oTObserved O: { }
. . .
aij
. . .
. . . . . .
bi(ot) bj(ot+1)
o1:t ot+2:T
ξt(i, j):Probability to be in state si at t, and si+1 at t + 1
Let oi:j denote the sequence of j − i + 1 observations (oi, oi+1, ..., oj−1, oj).
λ is ommitted since the model is a given for all the probabilities presented.
αt(i) =P(o1:t, qt = si) Backward prob of observations 1...t and state j after
βt+1(j) =P(ot+2:T |qt+1 = sj) Forward prob of observations t+2...T given state j
To decompose ξ into a product of backward, forward and transition probabilities,
we use a combination of Bayes theorem (1,4,6) and d-separation (5,8).
ξt(i, j) = P(qt = si, qt+1 = sj|O) =
P(qt = si, qt+1 = sj, O)
P(O)
(1)
P(O)ξt(i, j) = P(qt = si, qt+1 = sj, O) = P(qt = si, qt+1 = sj, o1:t, ot+1:T ) (2)
P(O)ξt(i, j) = P(qt = si, qt+1 = sj, o1:t, ot+1:T ) (3)
P(O)ξt(i, j) = P(o1:t, qt = si , ot+1:T , qt+1 = sj) (4)
P(O)ξt(i, j) = P(o1:t, qt = si)
αt(i)
P(ot+1:T , qt+1 = sj|o1:t, qt = si) (5)
P(O)ξt(i, j) = αt(i)P(ot+1:T , qt+1 = sj|qt = si) (6)
P(O)ξt(i, j) = αt(i)P(ot+1:T |qt+1 = sj, qt = si)P(qt+1 = sj|qt = si)
aij
(7)
P(O)ξt(i, j) = αt(i)aijP(ot+1, ot+2:T |qt+1 = sj, qt = si) (8)
P(O)ξt(i, j) = αt(i)aijP(ot+1|qt+1 = sj)
bj(ot+1)
P(ot+2:T |qt+1 = sj)
βt+1(j)
(9)
ξt(i, j) =
αt(i)aijbj(ot+1)βt+1(j)
P(O|λ)
(10)
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2. Nidhal Selmi,
Tanishka Singh
Hidden Markov Models
common probabilities
nselmi@asu.edu
Spring 2019
βt(i) Forward probability recursively in terms of βt+1(j)
βt(i) = P(ot+1:T |qt = si)
βt(i) = P(ot+1, ot+2:T |qt = si) (separate ot+1 given qt)
βt(i) = P(ot+1|qt = si)P(ot+2:T |qt = si)
Marginalizing over all next possible states j (with transitions and observations) given state i.
βt(i) =
N
j=1
P(qt+1 = sj|qt = si)P(ot+1|qt+1 = sj)P(ot+2:T |qt+1 = sj)
βt(i) =
N
j=1
aijbj(ot+1)βt+1(j)
γt(j):Probability to be in state sj given observations O
γt(j) = P(qt = sj|O) =
P(qt = sj, O)
P(O)
(Bayes)
P(O)γt(j) = P(qt = sj, O)
P(O)γt(j) = P(qt = sj, o1:t, ot+1:T ) (split O)
P(O)γt(j) = P(o1:t, qt = sj, ot+1:T )
P(O)γt(j) = P(o1:t, qt = sj)P(ot+1:T |o1:t, qt = sj) (Bayes)
P(O)γt(j) = P(o1:t, qt = sj)P(ot+1:T |qt = sj) (separate o1:t)
γt(j) =
αt(j)βt(j)
P(O|λ)
References
[1] L. R. Rabiner, ”A tutorial on hidden Markov models and selected applications in speech
recognition,” in Proceedings of the IEEE, vol. 77, no. 2, pp. 257-286, Feb. 1989.
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