Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariSpecifying a Gaussian Process!• Key point about Gaussian Stochastic Processes!– Joint distribution over N variables y 1 ,.., y N is completelyspecified by the second-order statistics,!• i.e., mean and covariance!• With mean zero, it is completely specified bycovariance of y(x) evaluated at any two values of xwhich is given by a kernel function!E[y(x n ) y(x m )] = k(x n ,x m )• For the Gaussian Process defined by the linearregression model y(x,w)=w T φ (x) withprior p(w) = N(w|0,α -1 I) the kernel function is38K nm = k (x n ,x m ) = (1/α) φ(x n ) T φ (x m )!
Machine Learning ! ! ! ! !SrihariDefining a Kernel Directly!• GP can also be specified directly bychoice of kernel function (instead ofindirectly by basis function)!• Samples of functions drawn fromGaussian processes for two differentchoices of kernel functions are shown next39
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Bayesian Linear Regression - CEDAR

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