Bayesian Decision Theory - gatech.edu

Bayesian Decision TheoryChapter2 (Duda, Hart & Stork)CS 7616 -Pattern RecognitionHenrik I ChristensenGeorgia Tech. Bayesian Decision Theory Design classifiers to recommend decisionsthat minimize some total expected risk . The simplest riskis the classification error ( , costs are equal). Typically, the riskincludes the costassociated with different State of nature (random variable): , 1for sea bass, 2for salmon Probabilities P( 1)and P( 2)(priors): , prior knowledge of how likely is to get a sea bass or a salmon Probability density function p(x) (evidence): , how frequently we will measure a pattern with feature value x( , xcorresponds to lightness) Terminology (cont d) Conditional probability density p(x/ j)(likelihood) : , how frequently we will measure a pattern with feature value xgiven that the pattern belongs to class , lightness distributionsbetween salmon/sea-basspopulationsTerminology (cont d) Conditional probability P( j /x) (posterior) : , the probability that the fish belongs to class jgiven measurement Rule Using Prior ProbabilitiesDecide 1ifP( 1)> P( 2);otherwise decide 2orP(error) = min[P( 1), P( 2)] Favours the most likely class.

Receiver Operating Characteristic (ROC) Curve • Every classifier employs some kind of a threshold. • Changing the threshold affects the performance of the system. • ROC curves can help us evaluate system performance for differentthresholds. θω ω a = PP()/() 21 212 22 121 11 ()( ) b ()( ) P P ωλ λ θ ωλ λ − = −

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  Operating, Receiver, Characteristics, Curves, Receiver operating characteristic, Roc curve

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