Je bilmes ee596awinter 2013dgms lecture 4 jan 23rd

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Unformatted text preview: are trees. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-27 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and elimination - forward recursion HMM forward recursion is just the elimination algorithm on the graph Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-28 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and elimination - forward recursion HMM forward recursion is just the elimination algorithm on the graph Choose elimination order: X1 , X2 , Q1 , X3 , Q2 , X4 , Q3 , X5 , . . . , XT , QT −1 , QT . Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-28 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and elimination - forward recursion HMM forward recursion is just the elimination algorithm on the graph Choose elimination order: X1 , X2 , Q1 , X3 , Q2 , X4 , Q3 , X5 , . . . , XT , QT −1 , QT . Evidence from delta functions, i.e., x1 = x1 ⇒ δ (x1 , x1 ), ¯ ¯ x2 = x2 ⇒ δ (x2 , x2 ), and so on. ¯ ¯ Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-28 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and elimination - forward recursion HMM forward recursion is just the elimination algorithm on the graph Choose elimination order: X1 , X2 , Q1 , X3 , Q2 , X4 , Q3 , X5 , . . . , XT , QT −1 , QT . Evidence from delta functions, i.e., x1 = x1 ⇒ δ (x1 , x1 ), ¯ ¯ x2 = x2 ⇒ δ (x2 , x2 ), and so on. ¯ ¯ We get: T ... p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ x4 q2 x3 q1 x2 x1 t=1 T = ... p(x1 |q1 )δ (x1 , x1 )p(q1 ) ¯ p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ x3 q1 x2 t=2 x1 ∆ p(¯1 |q1 )p(q1 )=α1 (q1 ) x Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-28 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and elimination - forward recursion T ... p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ x3 q1 p(x2 |q2 )p(q2 |q1 )δ (x2 , x2 )α(q1 ) ¯ x2 t=3 T = ... p(q2 |q1 )p(¯2 |q2 )α(q1 ) x p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ x4 q2 x3 q1 t=3 α2 (q2 ) = ... T =. . . p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ xr+2 qr xr+1 t=r+1 p(qr |qr−1 )p(¯r |qr )α(qr−1 ) x qr−1 αr (qr ) α-recursion becomes αt+1 (j ) = αt (i)p(Qt+1 = j |Qt = i)p(¯t+1 |Qt+1 = j ) x (4.20) i α1 (j ) = p(Q1 = j )p(¯1 |Q1 = j ) x Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 (4.21) page 4-29 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and elimination - forward recursion From the last line of the elimination (when we sum out qr−1 , we see α-recursion, which is: αt+1 (j ) = αt (i)p(Qt+1 = j |Qt = i)p(xt+1 |Qt+1 = j ) (4.22) α1 (j ) = p(Q1 = j )p(¯1 |Q1 = j ) x (4.23) i and Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-30 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do...
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This document was uploaded on 04/05/2014.

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