# UB CSE 555 - Bayes Decision Theory (21 pages)

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## Bayes Decision Theory

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- Pages:
- 21
- School:
- University at Buffalo, The State University of New York
- Course:
- Cse 555 - Introduction to Pattern Recognition

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Bayes Decision Theory Bayes Error Rate and Error Bounds Receiver Operating Characteristics Discrete Features Missing Features CSE 555 Srihari 0 Example of Bayes Decision Boundary Two Gaussian distributions each with four data points x2 10 3 1 6 3 2 2 1 2 2 2 4 6 8 x1 2 1 2 0 1 0 2 2 0 2 0 2 Inverse Matrices 2 0 1 0 1 2 0 1 1 2 2 0 1 2 1 x2 3 514 1 125 x1 0 1865 x1 Decision Boundary assumes P 1 P 2 0 5 CSE 555 Srihari 2 1 Bayes Error Rate Two Class Case optimal Stated for the multidimensional case regions R1 and R2 not easy to specify Multi Class Case Components of P error for equal priors and non optimal decision point x CSE 555 Srihari 2 Error Bounds for Normal Densities Evaluation of Error Integrals difficult Discontinuous decision regions Instead obtain bounds on error rate Useful inequality in obtaining a bound Minimum of two integers is smaller than the square root of their product more generally 1 min a b a b Proof for a b 0 and 0 1 If a b then a b 1 or a b b b or a b1 b CSE 555 Srihari 3 Chernoff Bound P error P 1 P1 2 p x 1 p1 x 2 dx for 0 1 Note that integral is over all space no need to impose integration limits If conditional probabilities are normal the integral can be evaluated analytically yielding p x 1 p 1 x 2 dx e k CSE 555 Srihari 4 Bhattacharya Bound Special case of Chernoff bound where 0 5 CSE 555 Srihari 5 Bhattacharya versus Chernoff Error Bounds as value of is varied Chernoff bound is never looser than the Bhattacharya bound Here Chernoff bound is at 0 66 and is slightly tighter than the Bhattacharya bound 0 5 CSE 555 Srihari 6 Example of Bhattacharya bound with normal densities x2 10 3 1 6 1 2 2 2 4 6 8 x1 3 2 2 1 2 0 1 0 2 2 0 2 0 2 2 k 1 2 4 06 P error 0 0087 CSE 555 Srihari 7 Receiver Operating Characteristics Distance between Gaussian distributions useful in experimental psychology radar detection medical diagnosis Interested in detecting a weak pulse or dim flash of light Detector detects a signal whose mean value is 1 when signal is

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