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**Unformatted text preview: **Machine Learning Srihari 1 Decision Theory Sargur Srihari srihari@cedar.buffalo.edu Machine Learning Srihari 2 Decision Theory Using probability theory to make optimal decisions Input vector x , target vector t Regression: t is continuous Classification: t will consist of class labels Summary of uncertainty associated is given by p( x,t ) Inference problem is to obtain p( x,t ) from data Decision: make specific prediction for value of t and take specific actions based on t Machine Learning Srihari 3 Medical Diagnosis Problem X-ray image of patient Whether patient has cancer or not Input vector x is set of pixel intensities Output variable t represents whether cancer or not C 1 is cancer and C 2 is absence of cancer General inference problem is to determine p(x,C k ) which gives most complete description of situation In the end we need to decide whether to give treatment or not. Decision theory helps do this Machine Learning Srihari 4 Bayes Decision How do probabilities play a role in making a decision? Given input x and classes C k using Bayes theorem Quantities in Bayes theorem can be obtained from p(x,C k ) either by marginalizing or conditioning wrt appropriate variable Machine Learning Srihari 5 Minimizing Expected Error Probability of mistake (2-class) Minimum error decision rule For a given x choose class for which integrand is smaller Since p( x ,C k )=p(C k | x )p( x ), choose class for which a posteriori...

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