Bayesian Statistics

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# Bayesian Statistics - PowerPoint PPT Presentation

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

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### Bayesian Statistics

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

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?

4 million – HIV-

1,400 – HIV+

3,960,000-

40,000+

14-

1,386+

3,960,000-

14-

40,000+

1,386+

P(HIV+|Test+) = 1,386/ (40,000 + 1,386)

= 3.35%

P(HIV+|Test-) = 14/ (3,960,000 + 14)

= 3.5x10-4%

Before the test

P(HIV+) = 1,400 / (1,400 + 4,000,000)

= 0.035%

P(HIV+) – Hypothesis (hidden) = 0.03%

P(HIV+|Test+)

P(Test+|HIV+)

99%

what we want

but is hard to

get to

P(Data) - data (observed)

P(Hyp) – Hypothesis (hidden)

P(Hyp|Data)

P(Data|Hyp)

easy to reason

what we want

but is hard to

get to

P(Data) - data (observed)

What is Bayes’ rule

Model

Prior

P(Data|Hyp) P(Hyp)

P(Hyp|Data) =

∑P(Data|H’) P(H’)

Normalization

P(Test+|HIV+) P(HIV+)

P(Data|Hyp) P(Hyp)

99% x1,400/(1,400 + 4,000,000)

99% x1,400

1,386

P(Hyp|Data) =

P(HIV+|Test+) =

P(HIV+|Test+) =

∑ P(Data|H’) P(H’)

99% x1,400/(1,400 + 4,000,000)+ 1% x4,000,000/(1,400 + 4,000,000)

1,386+ 40,000

P(Test+|HIV+) P(HIV+)+P(Test+|HIV-) P(HIV-)

99% x1,400+ 1% x4,000,000

=

=

=

3.3%

P(Hyp)

HIV+ 0.035%

HIV- 99.965%

P(Data|Hyp)

Data

Hyp Test- Test+

HIV- 99% 1%

HIV+ 1% 99%

P(Test+|HIV+) P(HIV+)

P(Data|Hyp) P(Hyp)

99% x 0.035%

0.0346%

0.0346%

P(Hyp|Data) =

P(HIV+|Test+) =

P(HIV+|Test+) =

∑ P(Data|H’) P(H’)

1.034%

99% x 0.035%+ 1% x 99.965%

P(Test+|HIV+) P(HIV+)+P(Test+|HIV-) P(HIV-)

0.0346% + 0.99965%

=

=

=

3.35%

P(Data|Hyp) P(Hyp)

P(Data|Hyp) P(Hyp)

P(Data)=∑ P(Data|H’) P(H’)

P(Hyp|Data) =

P(Hyp|Data) =

∑P(Data|H’) P(H’)

P(Data)

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

P(Hyp)

A 99.9%

C 0.1%

Reference

A

P(Data|Hyp)

Data

Hyp A C

A 99% 1%

C 1% 99%

C

Reference

A

C

C

A 99.9%

C 0.1%

A -> A 98.9%

A->C 0.999%

10-3%

C -> C 0.099%

A->C 0.999%

C -> C 0.099%

A->C 91%

C -> C 9%

A->C -> A

A->C->C 0.91%

C->C->A

C->C->C 8.9%

A->C->C 0.91%

C->C->C 8.9%

A->C->C 9.25%

C->C->C 90.75%

P(Data|Hyp) P(Hyp)=

P(Hyp) P(D1|Hyp) P(D2|Hyp)…P(Dn|Hyp)

P(Hyp)

AA 99.9%

AC 0.075%

CC 0.025%

P(Data|Hyp)

Data

Hyp A C

AA 99% 1%

AC 50% 50%

CC 1% 99%

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