Lecture5

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Unformatted text preview: to be generatively accurate, we can derive necessary conditions on the HMM, e.g., number of required states. Recall, nth -order Markov chain convertable to 1st-order one. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-73 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy For an HMM to be generatively accurate, we can derive necessary conditions on the HMM, e.g., number of required states. Recall, nth -order Markov chain convertable to 1st-order one. if D(p(x1:T )||ph (x1:T )) = 0, then the following mutual information quantities must be equal (5.62) I (XS1 ; XS2 ) = Ih (XS1 ; XS2 ) and where S1 , S2 ⊆ 1 : T , and where I (·; ·) is true mutual information, and Ih (·; ·) is the mutual information under the HMM Ih (XS1 ; XS2 ) = ph (xS1 , xS2 ) log xS1 ∪S2 Prof. Jeff Bilmes ph (xS1 , xS2 ) ph (xS1 )ph (xS2 ) EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page (5.63) 5-73 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy For an HMM to be generatively accurate, we can derive necessary conditions on the HMM, e.g., number of required states. Recall, nth -order Markov chain convertable to 1st-order one. if D(p(x1:T )||ph (x1:T )) = 0, then the following mutual information quantities must be equal (5.62) I (XS1 ; XS2 ) = Ih (XS1 ; XS2 ) and where S1 , S2 ⊆ 1 : T , and where I (·; ·) is true mutual information, and Ih (·; ·) is the mutual information under the HMM Ih (XS1 ; XS2 ) = ph (xS1 , xS2 ) log xS1 ∪S2 ph (xS1 , xS2 ) ph (xS1 )ph (xS2 ) (5.63) ∆ Define X¬t = {X1 , X2 , . . . , Xt−1 , Xt+1 , . . . , XT } (i.e., ¬t is set of all indices sans t). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-73 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy Theorem 5.8.1 (Necessary conditions for generative HMM accuracy.) An HMM with joint observation distribution ph (X1:T ) will accurately model the true distribution p(X1:T ) only if the following three conditions hold for all t: Ih (X¬t ; Qt ) ≥ I (Xt ; X¬t ), Ih (Qt ; Xt ) ≥ I (Xt ; X¬t ), and |DQ | ≥ 2I (Xt ;X¬t ) where Ih (X¬t ; Qt ) (resp. Ih (Qt ; Xt )) is the information transmission rate between X¬t and Qt (resp. Qt and Xt ) under an HMM, and I (Xt ; X¬t ) is the true information transmission rate between I (Xt ; X¬t ). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-74 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Proof: HMMs Generative Accuracy Proof. Accurate HMM (i.e., zero KL-divergence from true distribution) implies I (X¬t ; Xt ) = Ih (X¬t ; Xt ). We expand Ih (X¬t ; Qt , Xt ) in two ways using the chain rule of mutual information: (5.64) Ih (X¬t ; Qt , Xt ) = Ih (X¬t ; Qt ) + Ih (X¬t ; Xt |Qt ) (5.65) = I (X¬t ; Xt ) + Ih (X¬t ; Qt |Xt ) (5.67) = Ih (...
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This document was uploaded on 04/05/2014.

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