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Unformatted text preview: = qt+1
(B ) p(xt+2:T qt+1 )p(xt+1 qt+1 )p(qt+1 qt ) = (4.4) qt+1 where (A) follows from the chain rule probability, and (B) follows
since Xt+2:T ⊥ Xt+1 , Qt }Qt+1 and Xt+1 ⊥ Qt Qt+1 .
⊥{
⊥
Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 413 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM backwards recursion
This later assumption (Assumption II), however, is obligatory for the
beta or backward recursion in HMMs as we will now see.
p(xt+1,T qt )
p(qt+1 , xt+1 , xt+2:T qt ) =
qt+1
(A) p(xt+2:T qt+1 , xt+1 , qt )p(xt+1 qt+1 , qt )p(qt+1 qt ) = qt+1
(B ) p(xt+2:T qt+1 )p(xt+1 qt+1 )p(qt+1 qt ) = qt+1 where (A) follows from the chain rule probability, and (B) follows
since Xt+2:T ⊥ Xt+1 , Qt }Qt+1 and Xt+1 ⊥ Qt Qt+1 .
⊥{
⊥
Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 414 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM backwards recursion ∆ Using the deﬁnition βq (t) = p(xt+1:T Qt = q ), the above equations
imply the betarecursion
βq (t) = βr (t + 1)p(xt+1 Qt+1 = r)p(Qt+1 = rQt = q ), (4.5)
r and another expression for the full probability of evidence:
p(x1:T ) = βq (1)p(q )p(x1 q ). (4.6) q Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 415 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ p(x1:t ) via α and β
Recall, goal is to compute p(x1:T ), the probability of “evidence”,
eﬃciently. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 416 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ p(x1:t ) via α and β
Recall, goal is to compute p(x1:T ), the probability of “evidence”,
eﬃciently.
This probability may be computed using a combination of the alpha
and beta values at any t since
p(xt+1:T qt , x1:t )p(qt , x1:t ) p(qt , x1:t , xt+1:T ) = p(x1:T ) =
qt
(A) qt p(xt+1:T qt )p(qt , x1:t ) = = qt βqt (t)αqt (t)
qt where (A) follows since Xt+1:T ⊥ X1:t Qt in an HMM.
⊥ Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 416 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ p(x1:t ) via α and β
Recall, goal is to compute p(x1:T ), the probability of “evidence”,
eﬃciently.
This probability may be computed using a combination of the alpha
and beta values at any t since
p(xt+1:T qt , x1:t )p(qt , x1:t ) p(qt , x1:t , xt+1:T ) = p(x1:T ) = qt qt
(A) p(xt+1:T qt )p(qt , x1:t ) = = qt βqt (t)αqt (t)
qt where (A) follows since Xt+1:T ⊥ X1:t Qt in an HMM.
⊥
Note, above is true for any t. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 4  Jan 23rd, 2013 page 416 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM GM
Q1 Q2 X1 ... QT X2 Q1 Q2 XT X1 ... X2 QT XT For any p ∈ F (HMM, R), from the directed local Markov property,
we can immediately write down the joint as
T p(x1:T , q1:T ) = p(q1 )p(x1 q1 ) p(xt ...
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