t if we need this for one particular t this is

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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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