Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariGP for <strong>Regression</strong> (Direct Approach)!• We specify Gaussian Process directly over functions!– Abandon approach of defining a distribution over weights w• Take into account noise on observed target values as!! ! !t n = y n + ε n where y n =y (x n )– Noise process has a Gaussian distribution p(t n |y n )=N(t n |y n ,β -1 )• Note that target t n is output y n corrupted by noise• Defining t =(t 1 ,..,t N ) T our goal is to determine adistribution p(t)– Which is a distribution over functions
Machine Learning ! ! ! ! !SrihariGP for <strong>Regression</strong> (Direct Approach)!• Assuming noise is independent for each data point!!joint distribution of t =(t 1 ,..,t N ) T on values y = (y 1 ,..,y N ) T isp(t|y)=N(t|y,β -1 I N )• From definition of GP, marginal distribution of y is given by !!a Gaussian with zero mean, covariance matrix given by Gram matrix Kp(y) =N(y|0,K)• Where kernel function that determines K is chosen to express: !!property that for points x n , x m that are similar corresponding values y(x n ),y(x m ) will be more strongly correlated than for dissimilar points!!K depends on application!
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Bayesian Linear Regression - CEDAR

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