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Lecture 6. Bayes Rule

Lecture 6. Bayes Rule. David R. Merrell 90-786 Intermediate Empirical Methods for Public Policy and Management. AGENDA. Review Addition Law for Probability Multiplication Law for Probability Conditional Probability Bayes Rule Total Probability Rule Applications Interpretations.

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Lecture 6. Bayes Rule

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  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 ... Logic Set Arithmetic Simplification and x independence 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 A2 A4 B A1 A3

  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

  16. Next Time ... • Discrete Random Variables • Binomial Distribution • Poisson Distribution

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