9 pxt qt pxt1t qt pqt qt1 x1t1 prof je

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Unformatted text preview: ) (5.7) = p(xt |qt )βt (qt )p(qt |qt−1 , x1:t−1 )p(qt−1 , x1:t−1 ) (5.9) = p(xt |qt )p(xt+1:T |qt )p(qt , qt−1 , x1:t−1 ) (5.8) = p(xt |qt )βt (qt )p(qt |qt−1 )αt−1 (qt−1 ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 (5.10) page 5-37 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Edge Marginals So all of the edge marginals can be computed using the standard (α and β ) recursions. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-38 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Edge Marginals So all of the edge marginals can be computed using the standard (α and β ) recursions. There are several aspects of HMMs we will discuss: 1) Viterbi decoding/MPE, 2) sampling, 3) how flexible are HMMs, 4) real-world inference in HMMs (what to do when state space gets large), 5) time-space tradeoffs. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-38 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation (MPE) Crucial problem in HMMs is to solve MPE (also called Viterbi decoding): ∗ q1:T ∈ argmax p(¯1:T , q1:T ) x (5.11) q1:T ∈DQ1:T Note that computing the value of the max can be done just with an alternate to the α-recursion. Since max q1:T ∈DQ1:T p(¯1:T , q1:T ) x Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-39 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation (MPE) Crucial problem in HMMs is to solve MPE (also called Viterbi decoding): ∗ q1:T ∈ argmax p(¯1:T , q1:T ) x (5.11) q1:T ∈DQ1:T Note that computing the value of the max can be done just with an alternate to the α-recursion. Since max q1:T ∈DQ1:T p(¯1:T , q1:T ) = x Prof. Jeff Bilmes max q1:T ∈DQ1:T t p(¯t |qt )p(qt |qt−1 ) x EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 (5.12) page 5-39 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation (MPE) Crucial problem in HMMs is to solve MPE (also called Viterbi decoding): ∗ q1:T ∈ argmax p(¯1:T , q1:T ) x (5.11) q1:T ∈DQ1:T Note that computing the value of the max can be done just with an alternate to the α-recursion. Since max q1:T ∈DQ1:T p(¯1:T , q1:T ) = x max q1:T ∈DQ1:T t p(¯t |qt )p(qt |qt−1 ) x (5.12) = max p(xT |qT ) . . . max p(¯2 |q2 )p(q3 |q2 ) max p(¯1 |q1 )p(q2 |q1 ) x x qT q2 q1 (5.13) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-39 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Most Probable Explanation (MPE) Crucial problem in HMMs is to solve MPE (also called Viterbi decoding): ∗ q1:T ∈ argmax p(¯1:T , q1:T ) x (5.11) q1:T ∈DQ1:T Note that computing the value of the max can be done just with an alternate to the α-r...
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