# Je bilmes ee596awinter 2013dgms lecture 5 jan 25th

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Unformatted text preview: HMMs can do Summary Scratch Correlated &amp; Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (5.42) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (5.43) Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-60 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlated &amp; Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (5.42) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (5.43) X and Y are uncorrelated if cov(X, Y ) = 0. Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-60 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlated &amp; Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (5.42) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (5.43) 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 5 - Jan 25th, 2013 page 5-60 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlated &amp; Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (5.42) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (5.43) 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 5 - Jan 25th, 2013 page 5-60 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlated &amp; Covariance Correlation between two real random vectors X and Y cor(X, Y ) = E [XY ] (5.42) Covariance between two real random vectors X and Y cov(X, Y ) = E [(X − EX ) (Y − EY ) ] = E [XY ] − E [X ]E [Y ] (5.43) 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 5 - Jan 25th, 2013 page 5-60 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch Correlation over time of simple HMM Consider single-component Gaussian HMM (i.e., for each state, the observation distribution is a single multivariate Gaussian). Prof. Jeﬀ Bilmes EE596A/Winter 2013/DGMs – Lecture 5 - Jan 25th, 2013 page 5-61 (of 232) HMMs Trellis Other HMM queries MPE Sampling What HMMs can do Summary Scratch 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 currentl...
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## This document was uploaded on 04/05/2014.

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