58 covariance between two real random vectors x and y

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Unformatted text preview: (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 (4.57) Qt + 2 Ct + 2 Xt +2 EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-63 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlated & Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 (4.58) page 4-64 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlated & Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (4.58) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (4.59) Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-64 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlated & Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (4.58) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (4.59) X and Y are uncorrelated if cov(X, Y ) = 0. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-64 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlated & Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (4.58) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (4.59) X and Y are uncorrelated if cov(X, Y ) = 0. cov(X, Y ) = cor(X, Y ) if either the means are zero. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-64 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlated & Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (4.58) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (4.59) X and Y are uncorrelated if cov(X, Y ) = 0. cov(X, Y ) = cor(X, Y ) if either the means are zero. If X ⊥ Y then cov(X, Y ) = 0 but vice verse only if X, Y are jointly ⊥ Gaussian. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-64 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlated & Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (4.58) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (4.59) X and Y are uncorrelated if cov(X, Y ) = 0. cov(X, Y ) = cor(X, Y ) if either the means are zero. If X ⊥ Y then cov(X, Y ) = 0 but vice verse only if X, Y are jointly ⊥ Gaussian. cov(X, Y ) = 0 is indication of lack of linear dependence, and hint there might not be strong dependence at all. Prof. Jeff Bilmes EE596A/Winter 2013/DGMs – Lecture 4 - Jan 23rd, 2013 page 4-64 (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....
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This document was uploaded on 04/05/2014.

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