Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariLikelihood of Data• Assume noise precision parameter βt = y(x,w)+ε p(t|x,w,β)=N(t|y(x,w),β -1 )• Likelihood p(t|w) with Gaussian noise has anexponential formp(t | X, w, β) =N∏n=1N ( t n| w T φ(x n), β −1)This is the probability of data t given the parameters w andinput X6
Machine Learning ! ! ! ! !SrihariPosterior Distribution of Parameters• Posterior is proportional to Likelihood x Priorp(w | t) =– where likelihood function isp(t | X,w,β) =p(t | w)p(w)p(t)N ( t n| w T φ(x n),β −1)• Multiplying by Gaussian prior, p(w)=N(w|m 0 ,S 0 )posterior is also Gaussian, written directly asp(w|t)=N(w|m N ,S N )– Where m N is the mean of the posteriorgiven by m N = S N (S 0-1m 0 + βΦ T t)– And S N is the covariance matrix of posteriorgiven by S-1N = S-10 + β Φ T Φ N∏n =1w 1 Product of Gaussiansw 0 7
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Bayesian Linear Regression - CEDAR

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