Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariEquivalent Kernel• Posterior mean of w is m N =βS N Φ T t– where S N is the design matrix• Substitute mean value into <strong>Regression</strong> functiony(x,w) =M −1∑j= 0w jφ j(x) = w T φ(x)• Mean of predictive distribution at point x€y(x, m N) = m N T φ(x) = βφ(x) T S NΦ T t=N∑ βφ(x) T S Nφ(x n)t nn=1N∑= k(x,x n)t nn=1– Where k(x,x’)=βφ (x) T S N φ (x’) is the equivalent kernel• Mean of predictive distribution is a linear combinationof training set target variables
Machine Learning ! ! ! ! !SrihariKernel Function• <strong>Regression</strong> functions such asy(x, m N) =N∑n=1k(x,x n)t nk(x,x’)=βφ (x) T S N φ (x’)• That take a linear combination of thetraining set target values are known aslinear smoothers• They depend on the input values x n fromthe data set since they appear in thedefinition of S N 25
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Bayesian Linear Regression - CEDAR

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