Bayes Theorem Thomas R. Stewart, Ph.D. Center for Policy Research Rockefeller College of Public Affairs and Policy University at Albany State University.

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Presentation transcript:

Bayes Theorem Thomas R. Stewart, Ph.D. Center for Policy Research Rockefeller College of Public Affairs and Policy University at Albany State University of New York Public Administration and Policy PAD634 Judgment and Decision Making Behavior

PAD6342 A problem Outcome 1 Event P(Test result|Event) Test Outcome 2 Outcome 3 Outcome The data are organized like this: Outcome 1 Test P(Event|Test result) Event Outcome 2 Outcome 3 Outcome But to make a decision you need:

Bayes Theorem flips probabilities Conditional probability – P(A|X) means probability of A given X By symmetry: P(X|A)P(A) = P(A|X)P(X) Bayes theorem means “not A” See next two slides for expansion of P(X) in denominator. “A” is the event we are interested in, e.g., a disease. “X” is the evidence, e.g., a test result.

PAD6344 Expansion of P(A) X A Occurrence of X includes events “X and A” and “X and not A”

PAD6345 Expansion of P(A) for the denominator of Bayes Theorem Bayes theorem: Therefore…

PAD6346 Likelihood ratio form Arkes, H. R., & Mellers, B. A. (2002). Do juries meet our expectations? Law and Human Behavior, 26(6),

PAD6347 Bayesian belief updating p 0 is the prior probability of the event A Odds 0 = p 0 /(1-p 0 ) are the prior odds in favor of A lr X = P(X|A)/P(X|not A) is the likelihood ratio for new data X Odds 1 = [p 0 /(1-p 0 )]*lr X are the posterior odds in favor of A p 1 = Odds 1 /(1 + Odds 1 ) = posterior probability of event A = p(A|X) Belief updating spreadsheet

PAD6348 Dealing with causality Case 1 – X represents some data or evidence. It provides a clue to whether event A will occur (prediction) or has or is occurring (diagnosis). It is not causal. Examples – Doctor observes redness in the ear – Social worker observes unclean house.

PAD6349 Dealing with causality Case 2 – X represents some event that influences the likelihood that event A will occur. It has a causal influence Examples – Federal Reserve chairman says tax cut not a good idea. – Asian stock markets fall sharply.

PAD63410 What is easier to estimate? Both require prior probability of A Case 1 – Probability of X given A – Probability of X given not A Case 2 – Probability of A given X – Impact multiplier (Odds multiplier)