Transcription of Chapter 13 The Multivariate Gaussian
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Chapter 13 The Multivariate Gaussian
Chapter 13 The Multivariate GaussianIn this Chapter we present some basic facts regarding the Multivariate Gaussian discuss the two major parameterizations of the Multivariate Gaussian themomentparameterizationand thecanonical parameterization, and we show how the basic operationsof marginalization and conditioning are carried out in thesetwo parameterizations. We alsodiscuss maximum likelihood estimation for the Multivariate ParameterizationsThe Multivariate Gaussian distribution is commonly expressed interms of the parameters and , where is ann 1 vector and is ann n, symmetric matrix. (We will assumefor now that is also positive definite, but later on we will haveoccasion to relax thatconstraint).
As in the univariate case, the parameters µ and Σ have a probabilistic interpretation as the moments of the Gaussian distribution. In particular, we have the important result: µ = E(x) (13.2) T. (13.3) We will not bother to derive this standard result, but will provide a hint: diagonalize and appeal to the univariate case.
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