1. 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 T.STEWART@ALBANY.EDU Public Administration and Policy PAD634 Judgment and Decision Making Behavior

2. PAD6342 A problem Outcome 1 Event P(Test result|Event) Test Outcome 2 Outcome 3 Outcome 4 + - - - + + The data are organized like this: Outcome 1 Test P(Event|Test result) Event Outcome 2 Outcome 3 Outcome 4 + - - - + + But to make a decision you need:

3. 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.

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

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

6. PAD6346 Likelihood ratio form Arkes, H. R., & Mellers, B. A. (2002). Do juries meet our expectations? Law and Human Behavior, 26(6), 625-639.

7. 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

8. 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.

9. 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.

10. 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)