# Consider the way in which hmms are often used long

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Unformatted text preview: Jan 25th, 2013 page 5-58 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs stationarity depends on MC Therefore, the HMM’s stationarity condition is entirely determined by the stationarity condition of the underlying hidden Markov chain. Consider the way in which HMMs are often used: Long chains Chains with cycle transition matrices Chains with upper-triangular matrices Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-58 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs stationarity depends on MC Therefore, the HMM’s stationarity condition is entirely determined by the stationarity condition of the underlying hidden Markov chain. Consider the way in which HMMs are often used: Long chains Chains with cycle transition matrices Chains with upper-triangular matrices Chains with strictly left-to-right transitions. Ex: speech recognition Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-58 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch HMMs stationarity depends on MC Therefore, the HMM’s stationarity condition is entirely determined by the stationarity condition of the underlying hidden Markov chain. Consider the way in which HMMs are often used: Long chains Chains with cycle transition matrices Chains with upper-triangular matrices Chains with strictly left-to-right transitions. Ex: speech recognition Hence, in only rare cases, when HMMs are used, are they stationary stochastic processes. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-58 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Gaussian Mixture HMM One of the most widely used HMMs in practice is one where the observation distributions are Gaussian mixtures, where p(x|q ) = c = m p(x|q, c)p(c|q ) (5.39) N (x|µqm , Σqm ) cmq (5.40) and where N (x|µ, Σ) = Prof. Jeﬀ Bilmes 1 1 exp − (x − µ) Σ−1 (x − µ) d/2 2 |2π Σ| EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 (5.41) page 5-59 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Gaussian Mixture HMM One of the most widely used HMMs in practice is one where the observation distributions are Gaussian mixtures, where p(x|q ) = c = m p(x|q, c)p(c|q ) (5.39) N (x|µqm , Σqm ) cmq (5.40) and where 1 1 exp − (x − µ) Σ−1 (x − µ) d/2 2 |2π Σ| The HMM BN becomes N (x|µ, Σ) = Qt – 1 Ct – 1 Qt + 1 Ct + 1 Ct Xt –1 Prof. Jeﬀ Bilmes Qt Xt Xt +1 (5.41) Qt + 2 Ct + 2 Xt +2 EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-59 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlated & Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 (5.42) page 5-60 (of 232) HMMs Trellis Other HMM queries MPE Sampling What...
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## This document was uploaded on 04/05/2014.

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