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Unformatted text preview: (x − µ) Σ−1 (x − µ)
d/2
2
2π Σ
The HMM BN becomes
N (xµ, 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 463 (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 464 (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 464 (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 464 (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 464 (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 464 (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 464 (of 239) HMMs HMMs as GMs Other HMM queries What HMMs can do MPE Summ Correlation over time of simple HMM
Consider singlecomponent Gaussian HMM (i....
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This document was uploaded on 04/05/2014.
 Winter '14

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