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PATTERNRECOGNITIONANDMACHINELEARNINGCHAPTER1:INTRODUCTIONExampleHandwrittenDigitRecognitionPolynomialCurveFittingSum-of-SquaresErrorFunction0thOrderPolynomial1stOrderPolynomial3rdOrderPolynomial9thOrderPolynomialOver-fittingRoot-Mean-Square(RMS)Error:PolynomialCoefficientsDataSetSize:9thOrderPolynomialDataSetSize:9thOrderPolynomialRegularizationPenalizelargecoefficientvaluesRegularization:Regularization:Regularization:vs.PolynomialCoefficientsProbabilityTheoryApplesandOrangesProbabilityTheoryMarginalProbabilityJointProbabilityConditionalProbabilityProbabilityTheorySumRuleProductRuleTheRulesofProbabilitySumRuleProductRuleBayes’Theoremposteriorlikelihood×priorProbabilityDensitiesTransformedDensitiesExpectationsConditionalExpectation(discrete)ApproximateExpectation(discreteandcontinuous)VariancesandCovariancesTheGaussianDistributionGaussianMeanandVarianceTheMultivariateGaussianGaussianParameterEstimationLikelihoodfunctionMaximum(Log)LikelihoodPropertiesofandCurveFittingRe-visitedMaximumLikelihoodDeterminebyminimizingsum-of-squareserror,.PredictiveDistributionMAP:ASteptowardsBayesDeterminebyminimizingregularizedsum-of-squareserror,.BayesianCurveFittingBayesianPredictiveDistributionModelSelectionCross-ValidationCurseofDimensionalityCurseofDimensionalityPolynomialcurvefitting,M=3GaussianDensitiesinhigherdimensionsDecisionTheoryInferencestepDetermineeitheror.DecisionstepForgivenx,determineoptimalt.MinimumMisclassificationRateMinimumExpectedLossExample:classifymedicalimagesas‘cancer’or‘normal’DecisionTruthMinimumExpectedLossRegionsarechosentominimizeRejectOptionWhySeparateInferenceandDecision?•Minimizingrisk(lossmatrixmaychangeovertime)•Rejectoption•Unbalancedclasspriors•CombiningmodelsDecisionTheoryforRegressionInferencestepDetermine.DecisionstepForgivenx,makeoptimalprediction,y(x),fort.Lossf
