1. Lecture 6. Bayes Rule David R. Merrell 90-786 Intermediate Empirical Methods for Public Policy and Management

2. AGENDA Review Addition Law for Probability Multiplication Law for Probability Conditional Probability Bayes Rule Total Probability Rule Applications Interpretations

3. Addition Law for Probability P(A or B) = P(A) + P(B) - P(A and B) Example: A passed Exam 1 B passed Exam 2

4. If Mutually Exclusive... P(A or B) = P(A) + P(B) Note simplification of Addition Rule

5. Multiplication Law for Probability P(A and B) = P(A B) = P(A)P(B|A) = P(A|B)P(B) Example Prepared for Exam Passed Exam A B

6. If Independent... P(A and B) = P(A)P(B) Note simplification of Multiplication Rule

7. Some Connections... LogicSetArithmeticSimplification and xindependence or + mutually exclusive Note: independence is NOT mutual exclusivity

8. Conditional Probability Events A, B P(A and B) = P(B |A)P(A) = P(A|B)P(B) Definition:

9. Example--Conditional Probability

10. Bayes Rule P(A | B) = P(A) P(B | A) P(B) Proof: P(A and B) = P(A|B)P(B) = P(B|A)P(A)

11. Total Probability Rule A1A1 A2A2 B A3A3 A4A4

12. Example: Survey Sampling

13. Application of Bayes Rule: Weather Forecasting P(rain) =.3 P(likely | rain) =.95 P(unlikely | no rain) =.9

14. Interpretations of Bayes Rule Conditioning Flip Knowledge Change

15. AIDS Example: Excel Implementation HIV Screening for AIDS False Positives False Negatives

17. Next Time... Discrete Random Variables Binomial Distribution Poisson Distribution