3 qt1 b pxt2t qt1 pxt1 qt1 pqt1 qt 44 qt1 where

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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 4-13 (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 4-14 (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 beta-recursion βq (t) = βr (t + 1)p(xt+1 |Qt+1 = r)p(Qt+1 = r|Qt = 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 4-15 (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 4-16 (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 4-16 (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 4-16 (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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