Id joint probability under an hmm pxtth xtth th qtth

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Unformatted text preview: age 5-52 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Observations are not i.i.d. Joint probability under an HMM: p(Xt:t+h = xt:t+h ) t+h = qt:t+h j =t p(Xj = xj |Qj = qj )aqj −1 qj . Unless only one state is possible, observations do not factorize. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-52 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Observations are not i.i.d. Joint probability under an HMM: p(Xt:t+h = xt:t+h ) t+h = qt:t+h j =t p(Xj = xj |Qj = qj )aqj −1 qj . Unless only one state is possible, observations do not factorize. In an HMM, there are no statements of the form A⊥ B . ⊥ Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-52 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Observations are not i.i.d. Joint probability under an HMM: p(Xt:t+h = xt:t+h ) t+h = qt:t+h j =t p(Xj = xj |Qj = qj )aqj −1 qj . Unless only one state is possible, observations do not factorize. In an HMM, there are no statements of the form A⊥ B . ⊥ HMMs have conditional independence properties (like all DGMs have). p(Xt:t+h = xt:t+h |Qt:t+h = qt:t+h ) t+h = τ =t Prof. Jeff Bilmes p(Xτ = xτ |Qτ = qτ ). EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-52 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Observations are not “Viterbi i.i.d.” The Viterbi path (most-probable explanation) of an HMM is defined: ∗ q1:T ∈ argmax q1:T p(X1:T = x1:T , q1:T ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-53 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Observations are not “Viterbi i.i.d.” The Viterbi path (most-probable explanation) of an HMM is defined: ∗ q1:T ∈ argmax q1:T p(X1:T = x1:T , q1:T ) If we max-marginalize over the hidden states, does that lead to i.i.d. distribution? Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-53 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Observations are not “Viterbi i.i.d.” The Viterbi path (most-probable explanation) of an HMM is defined: ∗ q1:T ∈ argmax q1:T p(X1:T = x1:T , q1:T ) If we max-marginalize over the hidden states, does that lead to i.i.d. distribution? The “Viterbi” distribution of the HMM is: pvit (X1:T = x1:T ) = c p(X1:T = x1:T , Q1:T = (5.27) ∗ q1:T ) = c max p(X1:T = x1:T , Q1:T = q1:T ) q1:T T = c max q1:T t=1 p(Xt = xt |Qt = qt )p(Qt = qt |Qt−1 = qt−1 ) where c is a positive normalizing constant over x1:T . This is just a different semi-ring. The resulting distribution over observations does not in general factorize, so no i.i.d. here either. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-53 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs ca...
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This document was uploaded on 04/05/2014.

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