Bayesian Statistics

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

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

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?

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

P(Test+|HIV+)

P(HIV+|Test+)

P(HIV+) – Hypothesis (hidden) = 0.03%

P(Data) - data (observed)

what we wantbut is hard toget to

99%

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

What is Bayes’ rule

P(Data|Hyp) P(Hyp) P(Hyp|Data) =

AnswerNormalization

PriorModel

∑ P(Data|H’) P(H’)

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%

P(Data|Hyp)

DataHyp Test- Test+HIV- 99% 1%HIV+ 1% 99%

P(Hyp)

HIV+ 0.035%HIV- 99.965%

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%

Spreadsheet

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)

P(Data|Hyp)

DataHyp A CA 99% 1%C 1% 99%

P(Hyp)

A 99.9%C 0.1%

Reference A

C Read

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%

P(Data|Hyp) P(Hyp)=

P(Hyp) P(D1|Hyp) P(D2|Hyp)…P(Dn|Hyp)

Spreadsheet

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%

Spreadsheet

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