X prof je bilmes ee596awinter 2013dgms lecture 4 jan

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Unformatted text preview: 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 We have that α1 (Q1 = j ) = p(Q1 = j )p(¯1 |Q1 = j ) = p(¯1 , Q1 = j ), and x x α1 (q1 ) = p(¯1 , q1 ). x 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 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 We have that α1 (Q1 = j ) = p(Q1 = j )p(¯1 |Q1 = j ) = p(¯1 , Q1 = j ), and x x α1 (q1 ) = p(¯1 , q1 ). x Also, α2 (q2 ) = q1 p(q2 |q1 )p(¯2 |q2 )α(q1 ) = x x x ¯¯ x¯ q1 p(q1 , q2 , x1 , x2 ) = p(¯1 , x2 , q2 ) q1 p(q2 |q1 )p(¯2 |q2 )p(¯1 , q1 ) = In general, the forward (α) recursion has meaning αt (j ) = p(x1:t , Qt = j ) 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 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 We have that α1 (Q1 = j ) = p(Q1 = j )p(¯1 |Q1 = j ) = p(¯1 , Q1 = j ), and x x α1 (q1 ) = p(¯1 , q1 ). x Also, α2 (q2 ) = q1 p(q2 |q1 )p(¯2 |q2 )α(q1 ) = x x x ¯¯ x¯ q1 p(q1 , q2 , x1 , x2 ) = p(¯1 , x2 , q2 ) q1 p(q2 |q1 )p(¯2 |q2 )p(¯1 , q1 ) = In general, the forward (α) recursion has meaning αt (j ) = p(x1:t , Qt = j ) So α (forward) recursion is just an instance of the elimination algorithm run on the GM for the HMM graph. 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 MPE Summ HMMs and elimination - backward recursion Next, consider elimination order XT , QT , XT −1 , QT −1 , . . .. T .. p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ xT −2 qT −1 xT −1 qT xT t=1 T −1 =... p(xT |qT )p(qT |qT −1 )δ (xT , xT ) ¯ p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ xT −1 qT t=1 xT T −1 = ... p(xt |qt )p(qt |qt−1 )δ (xt , xt ) p(¯T |qT )p(qT |qT −1 ) ¯ x xT −1 qT Prof. Jeff Bilmes t=1 EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-31 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs and elimination - backward recursion T −1 ... p(xt |qt )p(qt |qt−1 )δ (xt , xt ) p(¯T |qT )p(qT |qT −1 ) ¯ x xT −2 qT t=1 1 βT (qT ) T −1 = ... p(xt |qt )p(qt |qt−1 )δ (xt , xt ) ¯ qT −1 xT −1 p(¯T |qT )p(qT |qT −1 )βT (qT ) x qT t=1 βT −1 (qT −1 ) This corresponds to the β recursion βt (qt ) = βt+1 (qt+1 )p(qt...
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This document was uploaded on 04/05/2014.

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