week5-bayes - Kemal Kılı Sabancı Üniversitesi Kemal...

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Unformatted text preview: Kemal Kılıç, Sabancı Üniversitesi Kemal Kılıç, Sabancı Üniversitesi Spring, 2010 Spring, 2010 Week 5 1 Decision Analysis MS 405 Modeling Uncertainty Kemal Kılıç Faculty of Engineering and Natural Sciences Spring, 2010 Spring, 2010 Probability: A Quick Introduction • Let A be a chance event • Probability of A: P(A) • P is a probability function that assigns a number in the range [0, 1] to each event in event space • Prior (a priori) probability of A, P(A): with no new information about A or related events (e.g., no patient information) • Posterior (a posteriori) probability of A: P(A) given certain (usually relevant) information (e.g., laboratory tests) Week 5 2 Spring, 2010 Spring, 2010 Probabilistic Calculus • If A, B are mutually exclusive : – P(A or B) = P(A) + P(B) • Thus: P(not(A)) = P(A c ) = 1-P(A) A B Week 5 3 Spring, 2010 Spring, 2010 Independence • In general P(A ∩ B) = P(A) * P(B|A) • A, B are independent iff – P(A ∩ B) = P(A) * P(B) – That is, P(A) = P(A|B) • If A,B are not mutually exclusive, but are independent: – P(A ∪ B) = 1-P(A c ∩ B c ) = 1-(1-P(A))*(1-P(B)) = P(A)+P(B)-P(A)*P(B) = P(A)+P(B) - P(A ∩ B) A B A ∩ B Week 5 4 Spring, 2010 Spring, 2010 Conditional Probability • Conditional probability : P(B|A) • Independence of A and B: P(B) = P(B|A) • Conditional independence of B and C, given A: iff P(B|A) = P(B|A ∩ C) – e.g., ice cream consumption, drownings given a specific season, say summer • P(A)=P(A ∩ B) + P(A ∩ B c ) (we can write in terms of conditional probabilities) ) ( ) | ( ) ( ) | ( ) (-- + = B P B A P B P B A P A P Week 5 5 Spring, 2010 Spring, 2010 Bayes Theorem = ) ( ) | ( ) ( ) | ( ) ( ) | ( ) | (-- + = B P B A P B P B A P B P B A P A B P ) ( ) | ( ) ( ) ( ) | ( B P B A P B A P A P A B P = ∩ = ) ( ) ( ) | ( ) | ( A P B P B A P A B P = Week 5 6 Spring, 2010 Spring, 2010 Bayes Theorem - Application T P D T P D P T D P positive test disease P ( ) ( | ) ( ) ( | ) ( : | ) + + = + = For example, for diagnostic purposes : • Suppose you have a data set of 2000 subjects some has a disease (say flue) and others do not. Suppose you made a test and determine the probabilities of subjects who test positive ( T+ ) among the ill and normal subjects. What is the probability of being ill given that you test positive? Week 5 7 Spring, 2010 Spring, 2010 Bayesian Statistics • The branch of statistics which is concerned with presentation and analysis of information with the incorporation of additional information, which typically reflects prior beliefs of the decision-maker • Bayesian Statistics has a small but loyal (and growing) following • Named after Thomas Bayes Week 5 8 Spring, 2010 Spring, 2010 Thomas Bayes • A Presbyterian minister and mathematician who constructed the theorem in an attempt to demonstrate the existence of God....
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week5-bayes - Kemal Kılı Sabancı Üniversitesi Kemal...

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