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Bayes Rule

Bayes Rule. Which is shorthand for:. Bayes' Rule. Product rule P(a  b) = P(a | b) P(b) = P(b | a) P(a)  Bayes' rule: P(a | b) = P(b | a) P(a) / P(b) or in distribution form P (Y|X) = P (X|Y) P (Y) / P (X) = α P (X|Y) P (Y)

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Bayes Rule

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  1. Bayes Rule Which is shorthand for: CS 3243 - Uncertainty

  2. Bayes' Rule • Product rule P(ab) = P(a | b) P(b) = P(b | a) P(a)  Bayes' rule: P(a | b) = P(b | a) P(a) / P(b) • or in distribution form P(Y|X) = P(X|Y) P(Y) / P(X) = αP(X|Y) P(Y) • Useful for assessing diagnostic probability from causal probability: • P(Cause|Effect) = P(Effect|Cause) P(Cause) / P(Effect) • E.g., let M be meningitis, S be stiff neck: P(m|s) = P(s|m) P(m) / P(s) = 0.5 × 0.0002 / 0.05 = 0.0002 • Note: posterior probability of meningitis still very small! CS 3243 - Uncertainty

  3. Bayes' Rule and conditional independence P(Cavity | toothache  catch) = α · P(toothache  catch | Cavity) P(Cavity) = α · P(toothache | Cavity) P(catch | Cavity) P(Cavity) • This is an example of a naïve Bayes model: P(Cause,Effect1, … ,Effectn) = P(Cause) πiP(Effecti|Cause) • Total number of parameters is linear in n CS 3243 - Uncertainty

  4. Naïve Bayes Classifier • Calculate most probable function value Vmap = argmax P(vj| a1,a2, … , an) = argmax P(a1,a2, … , an| vj) P(vj) P(a1,a2, … , an) = argmax P(a1,a2, … , an| vj) P(vj) Naïve assumption: P(a1,a2, … , an) = P(a1)P(a2) … P(an) CS 3243 - Uncertainty

  5. Naïve Bayes Algorithm NaïveBayesLearn(examples)For each target value vj P’(vj)← estimate P(vj) For each attribute value ai of each attribute a P’(ai|vj) ←estimate P(ai|vj) ClassfyingNewInstance(x)vnb= argmax P’(vj)ΠP’(ai|vj) ajε x vjεV CS 3243 - Uncertainty

  6. An Example (due to MIT’s open coursework slides) R1(1,1) = 1/5: fraction of all positive examples that have feature 1 = 1 R1(0,1) = 4/5: fraction of all positive examples that have feature 1 = 0 R1(1,0) = 5/5: fraction of all negative examples that have feature 1 = 1 R1(0,0) = 0/5: fraction of all negative examples that have feature 1 = 0 Continue calculation of R2(1,0) … CS 3243 - Uncertainty

  7. An Example (due to MIT’s open coursework slides) (1,1) (0,1) (1,0) (0,0) R1 = 1/5, 4/5, 5/5, 0/5 R2 = 1/5, 4/5, 2/5, 3/5 R3 = 4/5, 1/5, 1/5, 4/5 R4 = 2/5, 3/5, 4/5, 1/5 New x = <0, 0, 1, 1> S(1) = R1(0,1)*R2(0,1)*R3(1,1)*R4(1,1) = .205 S(0) = R1(0,0)*R2(0,0)*R3(1,0)*R4(1,0) = 0 S(1) > S(0), so predict v = 1. CS 3243 - Uncertainty

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