Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariEquivalent kernel for Gaussian BasisGaussian Basis ϕ(x)"φ j(x) = exp (x − µ j) 2 %$# 2s 2 '&Equivalent Kernelk(x,x’)=φ (x) T S N φ(x’)Plot of k(x,x’) shownas a functionof x and x’Peaks when x=x’For three values of xthe behaviorof k(x,x’) is shown as a slicexThey are localized around x, i.e., peaks when x =x’Local evidence is weighted more than distant evidenceKernel used directly in regressiony(x, m N) =N∑n=1k(x,x n)t nx’Data set used togenerate kernel were200 values ofx equally spacedin (-1,1)26
Machine Learning ! ! ! ! !SrihariEquivalent Kernel for Polynomial BasisFunctionφ j (x)=x jk(x,x’)=βφ (x) T S N φ (x’)Localized function of x’ even thoughcorresponding basis function is nonlocal27
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