Transcription of More on Multivariate Gaussians - Stanford University
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More on Multivariate Gaussians - Stanford University
More on Multivariate GaussiansChuong B. DoNovember 21, 2008Up to this point in class, you have seen Multivariate Gaussians arisein a number of appli-cations, such as the probabilistic interpretation of linear regression, gaussian discriminantanalysis, mixture of Gaussians clustering, and most recently, factor analysis. In these lec-ture notes, we attempt to demystify some of the fancier properties of Multivariate Gaussiansthat were introduced in the recent factor analysis lecture. The goal of these notes is to giveyou some intuition into where these properties come from, so that you can use them withconfidence on your homework (hint hint!) and DefinitionA vector-valued random variablex Rnis said to have amultivariate normal (or Gaus-sian) distributionwith mean Rnand covariance matrix Sn++1if its probabilitydensity function is given byp(x; , ) =1(2 )n/2| |1/2exp( 12(x )T 1(x )).
A vector-valued random variable x ∈ Rn is said to have a multivariate normal (or Gaus-sian) distribution with mean µ ∈ Rnn ++ 1 if its probability density function is given by p(x;µ,Σ) = 1 (2π)n/2|Σ|1/2 exp − 1 2 (x−µ)TΣ−1(x−µ) . 2 Gaussian facts Multivariate Gaussians turn out to be extremely handy in practice due to the ...
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