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Unformatted text preview: Kemal KÄ±lÄ±Ã§, SabancÄ± Ãœniversitesi Kemal KÄ±lÄ±Ã§, SabancÄ± Ãœniversitesi Spring, 2010 Spring, 2010 Week 5 1 Decision Analysis MS 405 Modeling Uncertainty Kemal KÄ±lÄ±Ã§ Faculty of Engineering and Natural Sciences Spring, 2010 Spring, 2010 Probability: A Quick Introduction â€¢ Let A be a chance event â€¢ Probability of A: P(A) â€¢ P is a probability function that assigns a number in the range [0, 1] to each event in event space â€¢ Prior (a priori) probability of A, P(A): with no new information about A or related events (e.g., no patient information) â€¢ 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 â€¢ If A, B are mutually exclusive : â€“ P(A or B) = P(A) + P(B) â€¢ Thus: P(not(A)) = P(A c ) = 1P(A) A B Week 5 3 Spring, 2010 Spring, 2010 Independence â€¢ In general P(A âˆ© B) = P(A) * P(BA) â€¢ A, B are independent iff â€“ P(A âˆ© B) = P(A) * P(B) â€“ That is, P(A) = P(AB) â€¢ If A,B are not mutually exclusive, but are independent: â€“ P(A âˆª B) = 1P(A c âˆ© B c ) = 1(1P(A))*(1P(B)) = P(A)+P(B)P(A)*P(B) = P(A)+P(B)  P(A âˆ© B) A B A âˆ© B Week 5 4 Spring, 2010 Spring, 2010 Conditional Probability â€¢ Conditional probability : P(BA) â€¢ Independence of A and B: P(B) = P(BA) â€¢ Conditional independence of B and C, given A: iff P(BA) = P(BA âˆ© C) â€“ e.g., ice cream consumption, drownings given a specific season, say summer â€¢ P(A)=P(A âˆ© B) + P(A âˆ© 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 = âˆ© = ) ( ) ( )  ( )  ( 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 : â€¢ 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 â€¢ 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 â€¢ Bayesian Statistics has a small but loyal (and growing) following â€¢ Named after Thomas Bayes Week 5 8 Spring, 2010 Spring, 2010 Thomas Bayes â€¢ 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, Conditional Probability, Bayesian probability, Thomas bayes

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