Bayesian Linear Regression - CEDAR

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Machine Learning ! ! ! ! !SrihariExample of Predictive Distribution• Data generated from sin(2πx)• Model: nine Gaussian basis fnsy(x,w) =w jφ j(x) = w T φ(x)• Predictive distribution8∑j=0p(t | x, t,α, β) = N(t | m N T ϕ(x),σ N 2 (x)) where σ N 2 (x) = 1 β + ϕ(x)T S Nϕ(x)Plot of p(t|x) for one data pointshowing mean (red)and one std dev (pink)Mean of Predictive Distribution⎛ φ0(x1)⎜⎜ φ0(x2)Φ = ⎜⎜⎝φ0(xN)m N =βS N Φ T tS N-1=αI+βΦ T Φφ (x11)...φφM −1M −1(x1)⎞⎟⎟⎟⎟(x )N ⎠20
Machine Learning ! ! ! ! !SrihariPredictive Distribution (Sinusoidal Data)p(t | x, t,α, β) = N(t | m N T ϕ(x),σ N 2 (x))N=1 N=2• Std dev of t issmallest inneighborhoodof data points• Uncertaintydecreases asmore datapoints areobservedMean of theGaussianPredictiveDistributionOne standarddeviationfrom MeanN=4 N=25Plot only shows point-wise predictive varianceTo show covariance between predictions at differentvalues of x draw samples from posterior distribution over wp(w|t) and draw samples of y(x,w)
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Bayesian Linear Regression - CEDAR

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