Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !Srihari<strong>Regression</strong>: Marginal Distribution!• From distributions p(t|y) and p(y) we can get marginal p(t)conditioned on input values x 1 ,..,x N• Applying Sum rule and product rule!p(t) = ∫ p(t | y)p(y)dy= N(t | 0,C)where covariance matrix C has the elements !C(x n ,x m ) = k(x n ,x m ) + β -1 δ nmDue to y(x)Due to ε From the result thatwhen p(y) and p(t|y) are Gaussianp(t) is also Gaussian• The two Gaussian sources of randomness, y(x) and ε areindependent, so their covariances simply add!δ specified by ε
Machine Learning ! ! ! ! !SrihariWidely used kernel function for GP!• Exponential of a quadratic form!!with addition of constant and linear terms!$k(x n,x m) = θ 0exp − θ 12 || x '%n− x m|| 2 (&) + θ 2+ θ 3x T nx m€Corresponds to a parametric model that isa linear function of input variables• Samples from this prior are plotted forvarious values of the parameters θ 0 , θ 1 , θ 2 ,and θ 345
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Bayesian Linear Regression - CEDAR

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