E for each state the observation distribution is a

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Unformatted text preview: e., for each state, the observation distribution is a single multivariate Gaussian). Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-65 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM Consider single-component Gaussian HMM (i.e., for each state, the observation distribution is a single multivariate Gaussian). Assume that the Markov chain is currently stationary, with stationary distribution π . Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-65 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM Consider single-component Gaussian HMM (i.e., for each state, the observation distribution is a single multivariate Gaussian). Assume that the Markov chain is currently stationary, with stationary distribution π . What about cov(Xt , Xt+h )? Is it zero? How does it decay? Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-65 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM Consider single-component Gaussian HMM (i.e., for each state, the observation distribution is a single multivariate Gaussian). Assume that the Markov chain is currently stationary, with stationary distribution π . What about cov(Xt , Xt+h )? Is it zero? How does it decay? Computing E [Xt ] EXt Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-65 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM Consider single-component Gaussian HMM (i.e., for each state, the observation distribution is a single multivariate Gaussian). Assume that the Markov chain is currently stationary, with stationary distribution π . What about cov(Xt , Xt+h )? Is it zero? How does it decay? Computing E [Xt ] EXt = Prof. Jeff Bilmes xp(Xt = x)dx EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-65 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM Consider single-component Gaussian HMM (i.e., for each state, the observation distribution is a single multivariate Gaussian). Assume that the Markov chain is currently stationary, with stationary distribution π . What about cov(Xt , Xt+h )? Is it zero? How does it decay? Computing E [Xt ] EXt = xp(Xt = x)dx = x p(Xt = x|Qt = i)πi dx (4.60) i Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-65 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM Consider single-component Gaussian HMM (i.e., for each state, the observation distribution is a single multivariate Gaussian). Assume that the Markov chain is currently stationary, with stationary distribution π . What about cov(Xt , Xt+h )? Is it zero? How does it decay? Computing E [Xt ] xp(Xt = x)dx = EXt = x p(Xt = x|Qt = i)πi dx (4.60) i E [Xt |Qt = i]πi...
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