Transcription of The Multivariate Gaussian Distribution
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The Multivariate Gaussian Distribution
The Multivariate Gaussian DistributionChuong B. DoOctober 10, 2008A vector-valued random variableX= X1 Xn Tis said to have amultivariatenormal (or Gaussian ) distributionwith mean Rnand covariance matrix Sn++1if its probability density function2is given byp(x; , ) =1(2 )n/2| |1/2exp 12(x )T 1(x ) .We write this asX N( , ). In these notes, we describe Multivariate Gaussians and someof their basic Relationship to univariate GaussiansRecall that the density function of aunivariate normal (or Gaussian ) distributionisgiven byp(x; , 2) =1 2 exp 12 2(x )2 .Here, the argument of the exponential function, 12 2(x )2, is a quadratic function of thevariablex. Furthermore, the parabola points downwards, as the coefficient of the quadraticterm is negative. The coefficient in front,1 2 , is a constant that does not depend onx;hence, we can think of it as simply a normalization factor used to ensure that1 2 Z exp 12 2(x )2 = from the section notes on linear algebra thatSn++is the space of symmetric positive definiten nmatrices, defined asSn++= A Rn n:A=ATandxTAx >0 for allx Rnsuch thatx6= 0.
The figure on the right shows a multivariate Gaussian density over two variables X1 and X2. In the case of the multivariate Gaussian density, the argument ofthe exponential function, −1 2 (x − µ)TΣ−1 definite, and since the inverse of any positive definite matrix is also positive definite, then for any non-zero vector z, zTΣ−1z ...
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