CS 59000 Statistical machine learningLecture 10Alan QiOutlineReview of Linear Regression with basis functionsRidge regression and lassoBayesian model selectionBayesian factor Empirical BayesianLinear Regression3Basis FunctionsExamples of Basis Functions (1)Maximum Likelihood Estimation (1)Maximum Likelihood Estimation (2)Sequential EstimationRegularized Least SquaresMore RegularizersVisualization of Regularized RegressionBayesian Linear RegressionPosterior Distributions of ParametersPredictive Posterior DistributionExamples of PredictiveDistributionQuestionSuppose we use Gaussian basis functions.What will happen to the predictive distribution if we evaluate it at places far from all training data points?Equivalent KernelGivenPredictive mean is thusEquivalent kernelBasis Function: Equivalent kernel:GaussianPolynomialSigmoidCovariance between two predictionsPredictive mean at nearby points will be highly correlated, whereas for more distant pairs of points the correlation will be smaller.Bayesian Model ComparisonSuppose we want to compare models .Given a training set , we computeModel evidence (also known as marginal likelihood):Bayes factor:Evidence and Parameter PosteriorMarginal likelihood and evidenceParameter posterior distribution and evidenceCrude Evidence ApproximationAssume posterior distribution is centered around its mode andEvidence penalizes over-complex modelsGiven M parametersMaximizing evidence leads to a natural trade-off between data fitting & model complexity.Evidence Approximation & Empirical BayesApproximating the predictive distribution by maximizing marginal likelihood. Where hyperparameters maximize the evidence .Known as Empirical Bayes or type II maximum likelihoodModel Evidence and Cross-ValidationRoot-mean-square error Model evidenceFitting polynomial regression
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