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Unformatted text preview: i EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-63 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Thus, cov(Xt , Xt+h ) is in general not equal to zero. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-64 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Thus, cov(Xt , Xt+h ) is in general not equal to zero. h But recall, Ah − 1π from lecture 3, and this is a matrix with all → rows equal to the stationary distribution. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-64 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Thus, cov(Xt , Xt+h ) is in general not equal to zero. h But recall, Ah − 1π from lecture 3, and this is a matrix with all → rows equal to the stationary distribution. Therefore, we have cov(Xt , Xt+h ) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 (5.54) page 5-64 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Thus, cov(Xt , Xt+h ) is in general not equal to zero. h But recall, Ah − 1π from lecture 3, and this is a matrix with all → rows equal to the stationary distribution. Therefore, we have cov(Xt , Xt+h ) = ij Prof. Jeﬀ Bilmes µi µj (Ah )ij πi − (5.54) µi π i i (5.55) µi π i i EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-64 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Thus, cov(Xt , Xt+h ) is in general not equal to zero. h But recall, Ah − 1π from lecture 3, and this is a matrix with all → rows equal to the stationary distribution. Therefore, we have cov(Xt , Xt+h ) = ij h − → Prof. Jeﬀ Bilmes ij (5.54) µi µj (Ah )ij πi − µi µj πj πi − µi π i i µi π i i (5.55) µi π i i µi π i (5.56) =0 i EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-64 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Thus, cov(Xt , Xt+h ) is in general not equal to zero. h But recall, Ah − 1π from lecture 3, and this is a matrix with all → rows equal to the stationary distribution. Therefore, we have cov(Xt , Xt+h ) = ij h − → ij (5.54) µi µj (Ah )ij πi − µi µj πj πi − µi π i i µi π i i (5.55) µi π i i µi π i (5.56) =0 i Thus, while the covariance between to observations is not necessarily zero in an HMM, once we are at a stationary distribution, this covariance goes to zero exponentially fast. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-64 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Thus, cov(Xt , Xt+h ) is in general not equal to zero. h But recall, Ah − 1π from lecture 3, and this is a matrix with all → rows equal to the stationary distribution. Therefore, we have cov(Xt ,...
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## This document was uploaded on 04/05/2014.

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