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Unformatted text preview: Kemal Kl, Sabanc niversitesi Kemal Kl, Sabanc niversitesi Spring, 2010 Spring, 2010 Week 5 1 Decision Analysis MS 405 Modeling Uncertainty Kemal Kl 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 ) = 1P(A) A B Week 5 3 Spring, 2010 Spring, 2010 Independence In general P(A B) = P(A) * P(BA) A, B are independent iff P(A B) = P(A) * P(B) That is, P(A) = P(AB) If A,B are not mutually exclusive, but are independent: P(A B) = 1P(A c B c ) = 1(1P(A))*(1P(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(BA) Independence of A and B: P(B) = P(BA) Conditional independence of B and C, given A: iff P(BA) = P(BA 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 decisionmaker 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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 Spring '11
 MajdabPadawnan
 Economics

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