人工智能课件AI2014S_Chap03ProbabilityDistributions幻灯片.pptxVIP

Artificial Intelligence Probability DistributionsDonghui WangAI Institute@ZJU2014.03ContentsThe Gaussian DistributionOther distributionsThe Exponential FamilyNonparametric MethodsReferences:Bishop. “Pattern Recognition and Machine Learning”, Chapter 2. 2006.Probability and Statistics Cookbook, /probability-and-statistics-cookbook//probability-and-statistics-cookbook//materials.html/materials.htmlThe Gaussian DistributionThe Gaussian DistributionSingle variable GaussianMultivariate GaussianThe Gaussian DistributionCentral limit theorem:Multivariate Gaussian Distributionmu = [0 0];Sigma = [.25 .3; .3 1];%Sigma = [.25 0; 0 1];%Sigma = [0.5 0; 0 0.5];x1 = -3:.1:3; x2 = -3:.1:3;[X1,X2] = meshgrid(x1,x2);F = mvnpdf([X1(:) X2(:)],mu,Sigma);F = reshape(F,length(x2),length(x1));surf(x1,x2,F);caxis([min(F(:))-.5*range(F(:)),max(F(:))]);axis([-3 3 -3 3 0 .4])xlabel(x1); ylabel(x2); zlabel(Probability Density);Mahalanobis distance Δ → Euclidean distanceMultivariate Gaussian DistributionThe matrix Σ can be taken to be symmetric, without loss of generality.the eigenvector equation for the covariance matrixMultivariate Gaussian DistributionJacobian factor or matrixMultivariate Gaussian DistributionIt’s normalized!Multivariate Gaussian DistributionExpectation of a random vector x:Multivariate Gaussian DistributionThe second order moments of the GaussianMultivariate Gaussian DistributionThe covariance of a random vector x:A general symmetric covariance matrix Σ will have D(D + 1)/2 independent parameters, and there are another D independent parameters in μ, giving D(D + 3)/2 parameters in total.2D independent parametersisotropic covariance, D + 1 independent parametersConditional Gaussian Distributionscovariance matrixprecision matrixcompleting the squareIf two sets of variables are jointly Gaussian, then the conditional distribution of one set conditioned on the other is again Gaussian.Completing the square:Marginal Gaussian Distributions：1. considering the terms involving xb and t