If we need this for one particular t this is

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Unformatted text preview: ter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-50 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ 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 4 - Jan 23rd, 2013 page 4-51 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ 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 q2 q3 q1 q2 q3 q4 q5 x1 x2 x3 x1 x2 x3 x4 x5 Involves re-sending messages (red). Each step requires O(T ) additional messages, in total O(T 2 ) computation. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-52 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Edge Marginals Need p(qt , qt+1 |x1:T ) for learning tasks Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-53 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Edge Marginals Need p(qt , qt+1 |x1:T ) for learning tasks Easy to obtain with both the α and β quantities, since: p(qt−1 , qt |x1:T ) = p(qt−1 , qt , x1:T ) p(x1:T ) (4.37) and p(qt−1 , qt , x1:T ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 (4.38) page 4-53 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Edge Marginals Need p(qt , qt+1 |x1:T ) for learning tasks Easy to obtain with both the α and β quantities, since: p(qt−1 , qt |x1:T ) = p(qt−1 , qt , x1:T ) p(x1:T ) (4.37) and p(qt−1 , qt , x1:T ) (4.38) = p(xt |qt )p(qt−1 , qt , x1:t−1 , xt+1:T ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 (4.39) page 4-53 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Edge Marginals Need p(qt , qt+1 |x1:T ) for learning tasks Easy to obtain with both the α and β quantities, since: p(qt−1 , qt |x1:T ) = p(qt−1 , qt , x1:T ) p(x1:T ) (4.37) and p(qt−1 , qt , x1:T ) (4.38) = p(xt |qt )p(qt−1 , qt , x1:t−1 , xt+1:T ) = p(xt |qt )p(xt+1:T |qt )p(qt , qt−1 , x1:t−1 ) Prof. Jeff Bilmes (4.39) (4.40) EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-53 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Edge Marginals Need p(qt , qt+1 |x1:T ) for learning tasks Easy to obtain with both the α and β quantities, since: p(qt−1 , qt |x1:T ) = p(qt−1 , qt , x1:T ) p(x1:T ) (4.37) and p(qt−1 , qt , x1:T ) (4.38) = p(xt |qt )p(qt−1 , qt , x1:t−1 , xt+1:T ) = p(xt |qt )p(xt+1:T |qt )p(qt , qt−1 , x1:t−1 ) (4.40) = p(xt |qt )βt (qt )p(qt |qt−1 , x1:t−1 )p(qt−1 , x1:t−1 ) Prof. Jeff Bilmes (4.39) (4.41) EE596A/Winter 2013/DGMs...
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