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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 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
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
If Mutually Exclusive ... P(A or B) = P(A) + P(B) Note simplification of Addition Rule
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
If Independent ... P(A and B) = P(A)P(B) Note simplification of Multiplication Rule
Some Connections ... Logic Set Arithmetic Simplification and x independence or + mutually exclusive Note: independence is NOT mutual exclusivity
Conditional Probability Events A, B P(A and B) = P(B |A)P(A) = P(A|B)P(B) Definition:
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)
Total Probability Rule A2 A4 B A1 A3
Application of Bayes Rule: Weather Forecasting P(rain) = .3 P(likely | rain) = .95 P(unlikely | no rain) = .9
Interpretations of Bayes Rule Conditioning Flip Knowledge Change
AIDS Example: Excel Implementation • HIV Screening for AIDS • False Positives • False Negatives
Next Time ... • Discrete Random Variables • Binomial Distribution • Poisson Distribution