CSE 150. Assignment 5
Out:
Tue Feb 15
Due:
Tue Feb 22
Recommended reading:
•
Russell & Norvig, Chapter 15.
•
L. R. Rabiner (1989). A tutorial on hidden Markov models and selected applications in speech recog
nition.
Proceedings of the IEEE
77(2):257–286.
5.1
Inference in HMMs
Consider a discrete HMM with hidden states
S
t
, observations
O
t
, transition matrix
a
ij
=
P
(
S
t
+1
=
j

S
t
=
i
)
and emission matrix
b
ik
=
P
(
O
t
=
k

S
t
=
i
)
. In class, we defined the forwardbackward probabilities:
α
it
=
P
(
o
1
, o
2
, . . . , o
t
, S
t
=
i
)
,
β
it
=
P
(
o
t
+1
, o
t
+2
, . . . , o
T

S
t
=
i
)
,
for a particular observation sequence
{
o
1
, o
2
, . . . , o
T
}
of length
T
. In terms of these probabilities, which
you may assume to be given, as well as the transition and emission matrices of the HMM, show how to
(efficiently) compute the posterior probability:
P
(
S
t

1
=
i, S
t
+1
=
k

S
t
=
j, o
1
, o
2
, . . . , o
T
)
.
In the above equation, you may assume that
t >
1
and
t < T
; in particular, you are
not
asked to consider
the boundary cases.
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 Spring '08
 Cottrell,G
 Speech recognition, Viterbi algorithm, Hidden Markov model, Markov models

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