Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariSamples from GaussianProcesses for two kernels!Functions are drawn !from !Gaussian processes!Note that each sample!is a function!Gaussian Kernel!k(x,x') = exp( − || x − x'|| 2 /2σ 2)Exponential Kernel!k(x,x') = exp( −θ | x − x'| )€€Ornstein-Uhlenbeck!process for !Brownian motion!40
Machine Learning ! ! ! ! !SrihariProb Distributions and Stochastic Processes!• A probability distribution describes the distributionsof scalars (x) and vectors (x)• A stochastic process describes distribution offunctions f(x)– Can think of a function as a very long vector!• Each entry in the vector is f (x) which is the value of function at x• A Gaussian process is a generalization of Gaussiandistributions where it is describing the distribution offunctions!• Since the vector is infinite-dimensional we constrain it toonly those points x corresponding to training and test points!41
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Bayesian Linear Regression - CEDAR

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