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UCLA STAT 231 - Lecture3-Decision(cont)

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3. Bayes Decision Theory: Part II.Bayes Decision Theory: Part IITwo-State CaseError Bounds:Chernoff and BhattaChernoff and BhattaMultiple SamplesSlide 8Probabilities of N SamplesError Rates for Large NROC curvesROC Curves:ROC CurvesROC Curves.Example: Boundary Detection 1.Example: Boundary Detection 2.Boundary Detection 3.Slide 18. SummaryLecture notes for Stat 231: Pattern Recognition and Machine Learning3. Bayes Decision Theory: Part II.Prof. A.L. YuilleStat 231. Fall 2004.Lecture notes for Stat 231: Pattern Recognition and Machine Learning Bayes Decision Theory: Part II1. Two-state case. Bounds for Risk.2. Multiple Samples.3. ROC curve and Signal Detection Theory.Lecture notes for Stat 231: Pattern Recognition and Machine LearningTwo-State CaseDetect state Let loss function pay a penalty of 1 for misclassification, 0 otherwise.Risk becomes Error. Bayes Risk becomes Bayes Error.Want to put bounds on the error.Lecture notes for Stat 231: Pattern Recognition and Machine LearningError Bounds:Use bounds to estimate errors. Bayes error:By We have:withLecture notes for Stat 231: Pattern Recognition and Machine LearningChernoff and Bhatta(I) the Bhattarcharyya boundwith Bhattarcharyya coefficient:(II) the Chernoff boundWith Chernoff Information:Lecture notes for Stat 231: Pattern Recognition and Machine Learning Chernoff and BhattaChernoff bound is tighter than the Bhatta bound.Both bounds are often good approximations – see Duda, Hart, Stork (pp 44, 48 example 1).There is also a lower bound:Bhatta and Chernoff will appear as exact errors rates for many samples.Lecture notes for Stat 231: Pattern Recognition and Machine Learning Multiple SamplesN Samples All from =A, or all from =B. (Bombers or Birds).Independence Assumption.Lecture notes for Stat 231: Pattern Recognition and Machine Learning Multiple SamplesPrior becomes unimportant for large N.Task becomes easier.Gaussian example:ThenwhereLecture notes for Stat 231: Pattern Recognition and Machine Learning Probabilities of N SamplesPosterior Distributions tend to Gaussians. (Central Limit Theorem). (Assumes independence or semi-independence).Results for N=0,1,2,3,50,200. (Left to Right, Top to Bottom).Lecture notes for Stat 231: Pattern Recognition and Machine LearningError Rates for Large NThe error rate E(N) decreases exponentially with the number N of samples.The Chernoff information:Recall for a single sample we have:Lecture notes for Stat 231: Pattern Recognition and Machine Learning ROC curvesReceiver Operator Characteristics (ROC) curves are more general than Bayes risk.Compare the performance of a human observer to Bayesian ideal for bright/dim light test.Suppose human does worse than Bayes risk--then maybe this is only decision bias.Lecture notes for Stat 231: Pattern Recognition and Machine LearningROC Curves:For two-state problems, the Bayes decision rule iswhere T depends on the priors and the loss function.The observer may use the correct log-likelihood ratio, but have the wrong threshold.E.g. the observer’s loss function may penalize false negatives (trigger-shy) or false positives (trigger-happy).Lecture notes for Stat 231: Pattern Recognition and Machine Learning ROC CurvesThe ROC curve plots the proportion of correct responses (hits) against the false positives as the threshold T changes.Requires altering the loss function of observers by rewards (chocolate) and penalties (electric shocks).The ROC curve gives information which is independent of the observer’s loss function.Lecture notes for Stat 231: Pattern Recognition and Machine Learning ROC Curves.Plot hits against false positives. For T large & +ve, bottom left of curve. T large & -ve, top right of curve. Tangent at 45 deg.s at T=0.Lecture notes for Stat 231: Pattern Recognition and Machine LearningExample: Boundary Detection 1.The boundaries of objects (right) usually occur where the image intensity gradient is large (left).Lecture notes for Stat 231: Pattern Recognition and Machine LearningExample: Boundary Detection 2.Learn the probability distributions for intensity gradient on and off labeled edges.Lecture notes for Stat 231: Pattern Recognition and Machine LearningBoundary Detection 3.Perform edge detection by log-likelihood ratio test.Lecture notes for Stat 231: Pattern Recognition and Machine LearningROC Curves:Special case: the likelihood functions are Gaussians with different means but same variance. Important in Psychology. See Duda, Hart, Stork.The Bayes Error can be computed from ROC curve.ROC curves distinguish between Discriminability and Decision Bias.Lecture notes for Stat 231: Pattern Recognition and Machine Learning. SummaryBounds on Error rates for single data. Bhatta and Chernoff Bounds.Multiple Samples. Error rates fall off exponentially with number of samples. Chernoff coefficient.ROC curves (Signal Detection


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UCLA STAT 231 - Lecture3-Decision(cont)

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