bayesian statistics n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Bayesian Statistics PowerPoint Presentation
Download Presentation
Bayesian Statistics

Loading in 2 Seconds...

play fullscreen
1 / 24

Bayesian Statistics - PowerPoint PPT Presentation


  • 132 Views
  • Uploaded on

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 ….

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Bayesian Statistics' - derex


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide5

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

slide7

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?

slide8

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%

slide9

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)

slide10

P(Hyp) – Hypothesis (hidden)

P(Hyp|Data)

P(Data|Hyp)

easy to reason

about

what we want

but is hard to

get to

P(Data) - data (observed)

what is bayes rule
What is Bayes’ rule

Model

Prior

P(Data|Hyp) P(Hyp)

P(Hyp|Data) =

∑P(Data|H’) P(H’)

Answer

Normalization

slide12

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%

slide13

P(Hyp)

HIV+ 0.035%

HIV- 99.965%

P(Data|Hyp)

Data

Hyp Test- Test+

HIV- 99% 1%

HIV+ 1% 99%

slide14

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%

slide16

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)

slide17

P(Hyp)

A 99.9%

C 0.1%

Reference

A

P(Data|Hyp)

Data

Hyp A C

A 99% 1%

C 1% 99%

Read

C

slide18

Reference

A

Read

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%

slide19

P(Data|Hyp) P(Hyp)=

P(Hyp) P(D1|Hyp) P(D2|Hyp)…P(Dn|Hyp)

slide21

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 statistics1
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