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Bayesian Statistics

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the theory that would not die how Bayes' rule cracked the enigma code, hunted down Russian submarines, and emerged triumphant from two centuries of controversy McGrayne, S. B., Yale University Press, 2011

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You are sitting in front of a doctor and she says …

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4 million – HIV- 1,400 – HIV+ Test has a 1% error rate If don’t have HIV then 1% of time it says you have it If you do have HIV then 1% of time it says you don’t have it You have been told that you have a positive test (and you don’t use intravenous drugs recreationally or partake of risky sexual practices) What is the probability that you actually have an HIV infection?

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4 million – HIV- 1,400 – HIV+ 3,960,000- 40,000+ 1,386+ 14- 3,960,000- 40,000+ 1,386+ 14- P(HIV+|Test+) = 1,386/ (40,000 + 1,386) = 3.35% P(HIV+|Test-) = 14/ (3,960,000 + 14) = 3.5x10 -4 % P(HIV+) = 1,400 / (1,400 + 4,000,000) = 0.035% Before the test

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P(Test+|HIV+) P(HIV+|Test+) P(HIV+) – Hypothesis (hidden) = 0.03% P(Data) - data (observed) what we want but is hard to get to 99%

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P(Data|Hyp) P(Hyp|Data) P(Hyp) – Hypothesis (hidden) P(Data) - data (observed) what we want but is hard to get to easy to reason about

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What is Bayes’ rule P(Data|Hyp) P(Hyp) P(Hyp|Data) = Answer Normalization Prior Model ∑ P(Data|H’) P(H’)

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P(Data|Hyp) P(Hyp) P(Hyp|Data) = ∑ P(Data|H’) P(H’) P(Test+|HIV+) P(HIV+) P(HIV+|Test+) = P(Test+|HIV+) P(HIV+)+P(Test+|HIV-) P(HIV-) 99% x1,400/(1,400 + 4,000,000) P(HIV+|Test+) = 99% x1,400/(1,400 + 4,000,000)+ 1% x4,000,000/(1,400 + 4,000,000) = 99% x1,400 99% x1,400+ 1% x4,000,000 1,386 1,386+ 40,000 = = 3.3%

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P(Data|Hyp) Data HypTest-Test+ HIV-99% 1% HIV+ 1%99% P(Hyp) HIV+ 0.035% HIV-99.965%

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P(Data|Hyp) P(Hyp) P(Hyp|Data) = ∑ P(Data|H’) P(H’) P(Test+|HIV+) P(HIV+) P(HIV+|Test+) = P(Test+|HIV+) P(HIV+)+P(Test+|HIV-) P(HIV-) 99% x 0.035% P(HIV+|Test+) = 99% x 0.035%+ 1% x 99.965% = 0.0346% 0.0346% + 0.99965% 0.0346% 1.034% = = 3.35%

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Spreadsheet

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P(Data|Hyp) P(Hyp) P(Hyp|Data) = ∑ P(Data|H’) P(H’) P(Data|Hyp) P(Hyp) P(Hyp|Data) = P(Data) P(Data)=∑ P(Data|H’) P(H’) P(Hyp|Data)P(Data)=P(Data|Hyp) P(Hyp)

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P(Data|Hyp) Data HypAC A99% 1% C 1%99% P(Hyp) A 99.9% C 0.1% Reference A C Read

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Reference A C A 99.9% C 0.1% A -> A 98.9% A->C 0.999% C -> C 0.099% 10 -3 % A->C 0.999% C -> C 0.099% Read C A->C 91% C -> C 9% A->C -> A C->C->A A->C->C 0.91% C->C->C 8.9% A->C->C 9.25% C->C->C 90.75% A->C->C 0.91%

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P(Data|Hyp) P(Hyp)= P(Hyp) P(D 1 |Hyp) P(D 2 |Hyp)…P(D n |Hyp)

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Spreadsheet

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P(Data|Hyp) Data HypAC AA99% 1% AC 50%50% CC 1%99% P(Hyp) AA 99.9% AC 0.075% CC 0.025%

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Spreadsheet

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Bayesian Statistics Simple mathematical basis Long period before it was used widely conceptual problems computationally difficult (Hyp can get very large) Technique useful for many otherwise intractable problems

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