For example the following left chain will have

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Unformatted text preview: 0.01 0.01 0.005 10 20 30 d 40 50 60 Other examples: very long chains, ladders, fixed-length distributions (histograms), and so on. HMMs can have flexible distributions, cost of extra states. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-69 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do - summary so far Observations are not i.i.d., but conditioned on state variables, they are independent. Observations are not “Viterbi i.i.d.” HMMs are a stationary process over p(x1:n ) whenever the underlying hidden Markov chain is a stationary process. Single Gaussian per state HMM: Covariance decays as: cov(Xt , Xt+h ) = ij h − → Prof. Jeff Bilmes ij µi µj (Ah )ij πi − µi µj πj πi − µi π i µi π i i i µi π i i µi π i =0 i EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-70 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch What HMMs can do - summary so far but mutual information (in practice) can apparently extend in time reasonably far (but also decays). Parameter sharing means enormous flexibility in state duration models (e.g., negative binomial, mixtures thereof, fixed histograms). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-71 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy We can view an HMM as an approximate generative distribution of the observation variables, as in ph (x1:T ) ≈ p(x1:T ) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-72 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy We can view an HMM as an approximate generative distribution of the observation variables, as in ph (x1:T ) ≈ p(x1:T ) Given that ph is an approximation, one that is a mixture ph (x1:T ) = (5.60) ph (x1:T , q1:T ) q1:T what can we say about ph and its accuracy? Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-72 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs Generative Accuracy We can view an HMM as an approximate generative distribution of the observation variables, as in ph (x1:T ) ≈ p(x1:T ) Given that ph is an approximation, one that is a mixture ph (x1:T ) = (5.60) ph (x1:T , q1:T ) q1:T what can we say about ph and its accuracy? Accuracy can be measured by KL-divergence D(p(x1:T )||ph (x1:T )) = p(x1:T ) log x1:T p(x1:T ) ph (x1:T ) (5.61) and if D(p(x1:T )||ph (x1:T )) = 0, then the HMM is perfectly generatively accurate. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-72 (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. 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...
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This document was uploaded on 04/05/2014.

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