Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariProperties of Posterior1. Since posterior p(w|t)=N(w|m N ,S N ) is Gaussianits mode coincides with its mean– Thus maximum posterior weight vector is w MAP = m N 2. For infinitely broad prior S 0 =α -1 I with α -> 0 – Then mean m N reduces to the maximum likelihoodvaluew ML= (Φ T Φ) −1 Φ T t3. If N = 0, posterior reverts to the prior4. If data points arrive sequentially, then posteriorto any stage acts as prior distribution forsubsequent data points8
Machine Learning ! ! ! ! !SrihariChoose a simple Gaussian prior p(w)• Zero mean (m 0 =0) isotropic• (same variances) Gaussianp(w |α ) = N(w | 0,α −1 I) Single precisionparameter• Corresponding posterior distribution isp(w|t)=N(w|m N ,S N )wherem N =βS N Φ T t and S N-1=αI+βΦ T ΦNote:β is noise precision andα is variance of parameter w in priorw 1 PointEstimatewithinfinitesamplesw 0 9
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Bayesian Linear Regression - CEDAR

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