Transcription of 1 Multivariate Normal Distribution - Princeton University
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1 Multivariate Normal Distribution - Princeton University
STA561: Probabilistic machine learningGaussian Models (9/9/13)Lecturer: Barbara EngelhardtScribes: Xi He, Jiangwei Pan, Ali Razeen, Animesh Srivastava1 Multivariate Normal DistributionThe Multivariate Normal Distribution (MVN), also known asmultivariate gaussian, is a generalization ofthe one-dimensional Normal Distribution to higher dimensions. The probability density function (pdf) of anMVN for a random vectorx Rdas follows:N(x| , ),1(2 )d/2| |1/2exp[ 12(x )T 1(x )](1)where =E[x] Rdis the mean vector, and =cov[x] isd dsymmetric positive definite matrix, knownas the covariance matrix. 1is known as the precision 1 21steigenvector of 2ndeigenvector of Figure 1: 2 dimensional Gaussian 1 shows a 2-dimensional Gaussian density. The random vectors span two dimensions and are denoted inthe plot byX1(x-axis) andX2(y-axis).
orthogonal to each other as shown in the Fig. 1. The rst eigenvalue is the direction of maximum variance ... similar projection, but considering only a slice of the space at the conditional random variable. When the o -diagonal elements of the covariance matrix are 0, the conditional distribution is identical to the marginal ...
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