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Statistics Workshop Bayes Theorem J-Term 2009 Bert Kritzer

Statistics Workshop Bayes Theorem J-Term 2009 Bert Kritzer. Bayesian Inference.

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Statistics Workshop Bayes Theorem J-Term 2009 Bert Kritzer

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  1. Statistics WorkshopBayesTheoremJ-Term 2009Bert Kritzer

  2. Bayesian Inference A method of using prior information about the probability of some event combined with conditional probabilities about consequences of that event to obtain post hoc probabilities using the actual observed consequences. We know the percentage of the population having some medical condition and we also know the probabilities of observing a symptom among those who do and those who do not have that condition. What is the probability that someone has the condition if we observe the symptom?

  3. Addition & Multiplication Rules Revisted As a single event As two events occurring together General case Special case If E and F are statistically independent If E and F are mutually exclusive

  4. Buses and Pedestrians 95% of buses are Metro Transit (MT) 80% of the time client correctly identify type P(B) = Probability of client identifying as MT P(A) = PriorProbability of bus being MT (.95) P(B|A) = Conditional probability of client saying it was MT give that it was in fact MT (.8) P(A|B) = Post hoc probability that it was in fact MT given that client says it was MT P(AB) = Joint probability that client says it was MT and it was in fact MT

  5. A A B P(AB) P(AB) P(AB) P(AB) B Combined Probabilities Conditional Joint P(A|B) P(B|A) P(AB) P(A|B) P(B|A) P(AB) P(A|B) P(B|A) P(AB) P(A|B) P(B|A) P(AB) A bus is Metro B client says bus is Metro

  6. Multiplication Rule Probability of saying it was MT:

  7. Summary

  8. Bayes Theorem

  9. P(MT given client says it’s MT) P(A) = .95 P of any bus being MT P(A)=.05 P of any bus not MT P(B|A) = .80 P of saying MT if it was MT P(B|A)=.20 P of saying MT if not MT

  10. Thinking Pictorially A A P(AB)=.76 P(AB)=.01 B B P(AB)=.04 P(AB)=.19

  11. P(not MT given client says not MT) P(A) = .95 P of any bus being MT P(A)=.05 P of any bus not MT P(B|A)= .80 P of saying not MT if not MT P(B|A)=.20 P of saying not MT if it is MT

  12. Client Says It Is Not MT A A P(AB)=.76 P(AB)=.01 B P(AB)=.04 P(AB)=.19 B

  13. .2 .19 ID-MT .8 ID-MT .04 Tree Trimming ID-MT .8 .76 MT .95 ID-MT .2 .05 .01 MT

  14. Online Bayes Calculator http://statpages.org/bayes.html

  15. Diagnosing Sexual Abuse

  16. 95% Accurate Diagnosis

  17. 20% Abuse Rate

  18. Different Error Rates

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