# Id joint probability under an hmm pxtth xtth th qtth

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 04/05/2014.

Ask a homework question - tutors are online