11 pxt qt pqt qt1 px1t1 qt1 412 qt1 qt1 if

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Unformatted text preview: , qt , qt−1 ) (4.11) p(xt |qt )p(qt |qt−1 )p(x1:t−1 , qt−1 ) (4.12) qt−1 = qt−1 ∆ If the following quantity is defined αq (t) = p(x1:t , Qt = q ), then the preceding equations imply that p(Qt = q |Qt−1 = r)αr (t − 1) αq (t) = p(xt |Qt = q ) (4.13) r Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-9 (of 239) Logistics Review HMM Forward Recursion Thus, p(x1:T ) = q αq (T ), and the entire computation requires only O(|DQ |2 T ) operations. To derive this recursion, it was necessary to use only the fact that Xt was independent of its past given Qt (call this Assumption I) — in an HMM, Xt is also independent of the future given Qt , but this was not yet used (call this Assumption II). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-10 (of 239) Logistics Review Scratch Paper Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-11 (of 239) Logistics Review Scratch Paper Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-12 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM backwards recursion This later assumption (Assumption II), however, is obligatory for the beta or backward recursion in HMMs as we will now see. p(xt+1:T |qt ) Prof. Jeff Bilmes (4.1) EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-13 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM backwards recursion This later assumption (Assumption II), however, is obligatory for the beta or backward recursion in HMMs as we will now see. p(xt+1:T |qt ) (4.1) p(qt+1 , xt+1 , xt+2:T |qt ) = (4.2) qt+1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-13 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM backwards recursion This later assumption (Assumption II), however, is obligatory for the beta or backward recursion in HMMs as we will now see. p(xt+1:T |qt ) (4.1) p(qt+1 , xt+1 , xt+2:T |qt ) = (4.2) qt+1 (A) p(xt+2:T |qt+1 , xt+1 , qt )p(xt+1 |qt+1 , qt )p(qt+1 |qt ) (4.3) = qt+1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-13 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM backwards recursion This later assumption (Assumption II), however, is obligatory for the beta or backward recursion in HMMs as we will now see. p(xt+1:T |qt ) (4.1) p(qt+1 , xt+1 , xt+2:T |qt ) = (4.2) qt+1 (A) p(xt+2:T |qt+1 , xt+1 , qt )p(xt+1 |qt+1 , qt )p(qt+1 |qt ) (4.3) = qt+1 (B ) p(xt+2:T |qt+1 )p(xt+1 |qt+1 )p(qt+1 |qt ) = (4.4) qt+1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-13 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM backwards recursion This later assumption (Assumption II), however, is obligatory for the beta or backward recursion in HMMs as we will now see. p(xt+1:T |qt ) (4.1) p(qt+1 , xt+1 , xt+2:T |qt ) = (4.2) qt+1 (A) p(xt+2:T |qt+1 , xt+1 , qt )p(xt+1 |qt+1 , qt )p(qt+1 |qt ) (4.3)...
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This document was uploaded on 04/05/2014.

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