Hmm as a junction tree various options for the

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Unformatted text preview: d, 2013 page 4-46 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMMs - summary so far Clique marginals are needed for standard learning procedures (EM and gradient calculation) Forward (α) recursion is elimination from left-to-right (equivalently, just LBP message passing) Backward (β ) recursion is elimination from right-to-left (again also LBP) “Inward” inference also possible via alternating elimination orders. HMM as a junction tree, various options for the cliques without increasing state space. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-47 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. γ ( qt ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) γ ( qt ) = qt+1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) = γ ( qt ) = qt+1 Prof. Jeff Bilmes p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) qt+1 EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) = γ ( qt ) = qt+1 p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) qt+1 p(qt |qt+1 , x1:T )γ (qt+1 ) = qt+1 Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) = γ ( qt ) = qt+1 p(qt |qt+1 , x1:T )γ (qt+1 ) = = qt+1 Prof. Jeff Bilmes p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HMM, other recursions It is possible to derive a temporal recursion for quantities other than α and β . E.g., here is a γt (j ) = p(Qt = j |x1:T ) backwards recursion. p(qt , qt+1 |x1:T ) = γ ( qt ) = qt+1 p(qt |qt+1 , x1:T )γ (qt+1 ) = = qt+1 = qt+1 Prof. Jeff Bilmes p(qt |qt+1 , x1:T )p(qt+1 |x1:T ) qt+1 p(qt |qt+1 , x1:t )γ (qt+1 ) qt+1 p(qt , qt+1 , x1:t ) γ (qt+1 ) qt p(qt , qt+1 , x1:t ) EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-48 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ HM...
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