bayesian statistics. the theory that would not die how bayes' rule cracked the enigma code,...
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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, andemerged triumphant from two centuries ofcontroversy
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 intravenousdrugs 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 wantbut is hard toget to
99%
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P(Data|Hyp)
P(Hyp|Data)
P(Hyp) – Hypothesis (hidden)
P(Data) - data (observed)
what we wantbut is hard toget to
easy to reason about
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What is Bayes’ rule
P(Data|Hyp) P(Hyp) P(Hyp|Data) =
AnswerNormalization
PriorModel
∑ 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)
DataHyp Test- 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)
DataHyp A CA 99% 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->AA->C->C 0.91% C->C->C 8.9%
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(D1|Hyp) P(D2|Hyp)…P(Dn|Hyp)
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Spreadsheet
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P(Data|Hyp)
DataHyp A CAA 99% 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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