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Unformatted text preview: ng What HMMs can do Summary Scratch Prediction p(qt |x1:s ), with t > s. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-34 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Prediction p(qt |x1:s ), with t > s. This is also easily obtained from the αs for we have p(qt , x1:s ) = qs ,qs+1 ,...,qt−1 = qs ,qs+1 ,...,qt−1 (5.2) p(qt , qt−1 , . . . , qs+1 , qs , x1:s ) p(qt |qt−1 )p(qt−1 |qt−2 ) . . . p(qs+1 |qs )αqs (s) (5.3) = qt−1 p(qt |qt−1 ) qt−2 (qt−1 |qt−2 ) · · · qs p(qs+1 |qs )αqs (s) (5.4) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-34 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Prediction p(qt |x1:s ), with t > s. This is also easily obtained from the αs for we have p(qt , x1:s ) = qs ,qs+1 ,...,qt−1 = qs ,qs+1 ,...,qt−1 (5.2) p(qt , qt−1 , . . . , qs+1 , qs , x1:s ) p(qt |qt−1 )p(qt−1 |qt−2 ) . . . p(qs+1 |qs )αqs (s) (5.3) = qt−1 p(qt |qt−1 ) qt−2 (qt−1 |qt−2 ) · · · qs p(qs+1 |qs )αqs (s) (5.4) And then we just normalize. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-34 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Prediction Thus, this is like running elimination (or message passing) on a graph where some of the observations are removed, as in: q1 q2 q3 x1 x2 q4 q5 x3 for computing p(q5 |x1:3 ). Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-35 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Smoothing p(qt |x1:u ), with t < u. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-36 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Smoothing p(qt |x1:u ), with t < u. If we need this for one particular t, this is identical to one of the posteriors we already have p(qt |x1:T ) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-36 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Smoothing p(qt |x1:u ), with t < u. If we need this for one particular t, this is identical to one of the posteriors we already have p(qt |x1:T ) Otherwise, we might need to update p(qt |x1:u ) for each t < u as u increases (observations come in). q1 q1 q2 q1 q2 q3 q4 x1 x1 x2 x1 x2 x3 x4 q1 q3 q1 q2 q3 q4 q5 x1 Prof. Jeﬀ Bilmes q2 x2 x3 x1 x2 x3 x4 x5 EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-36 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Smoothing p(qt |x1:u ), with t < u. If we need this for one particular t, this is identical to one of the posteriors we already have p(qt |x1:T ) Otherwise, we might need to update p(qt |x1:u ) for each t < u as u increases (observations come in). q1 q1 q2 q1 q2 q3 q4 x1 x1 x2 x1 x...
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## This document was uploaded on 04/05/2014.

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