Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariKernel Function formulation• Formulation of Linear Regression in terms ofkernel function suggests:– Directly define kernel functions and use to makepredictions for input x• Leads to Gaussian Processes for Regressionand Classification• Can be shown that• Equivalent kernel satisfies important property:– where€€k(x,z) =ψ(x) T ψ(z)ψ(x) = β 1/ 2 S N1/ 2 φ(x)N∑n=1k(x,x n) =130
Machine Learning ! ! ! ! !SrihariProbabilistic Linear Regression Revisited!• Re-derive the predictive distribution by workingin terms of distribution over functions y(x,w)– It will provide a specific example of a GaussianProcess• Consider model with M fixed basis functionsy(x)=w T φ (x)where x is the input vector and w is the M-dimensional weight vector• Assume a Gaussian distribution of weight wp(w) = N(w|0,α -1 I)• Probability distribution over w induces aprobability distribution over functions y(x)yx
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Bayesian Linear Regression - CEDAR

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