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Gov 51: Bayes Rule Matthew Blackwell Harvard University 1 / 8 QAnon You meet a man named Steve and he tells you that he is a Republican. You have been interested in meeting someone who believes in the QAnon conspiracy theory. Given what you

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SLIDE 1

Gov 51: Bayes Rule

Matthew Blackwell

Harvard University

1 / 8

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SLIDE 2

QAnon

You meet a man named Steve and he tells you that he is a Republican. You have been interested in meeting someone who believes in the QAnon conspiracy theory. Given what you know about Steve, would you guess that he believes in QAnon or not?

  • Common response: probably believes in QAnon since believers tend to

be Republicans.

  • Base rate fallacy: ignores how uncommon QAnon believers are!

2 / 8

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SLIDE 3

QAnon

You meet a man named Steve and he tells you that he is a Republican. You have been interested in meeting someone who believes in the QAnon conspiracy theory. Given what you know about Steve, would you guess that he believes in QAnon or not?

  • Common response: probably believes in QAnon since believers tend to

be Republicans.

  • Base rate fallacy: ignores how uncommon QAnon believers are!

2 / 8

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SLIDE 4

QAnon

You meet a man named Steve and he tells you that he is a Republican. You have been interested in meeting someone who believes in the QAnon conspiracy theory. Given what you know about Steve, would you guess that he believes in QAnon or not?

  • Common response: probably believes in QAnon since believers tend to

be Republicans.

  • Base rate fallacy: ignores how uncommon QAnon believers are!

2 / 8

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SLIDE 5

Visualizing QAnon support

Qanon nonbelievers

3 / 8

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SLIDE 6

Visualizing QAnon support

Qanon nonbelievers Qanon believers

3 / 8

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SLIDE 7

Visualizing QAnon support

Qanon nonbelievers Qanon believers Qanon Republicans

3 / 8

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SLIDE 8

Visualizing QAnon support

Qanon nonbelievers Qanon believers Qanon Republicans non-Qanon Republicans

3 / 8

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SLIDE 9

Visualizing QAnon support

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers Qanon Republicans non-Qanon Republicans

3 / 8

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SLIDE 10

Visualizing QAnon support

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers Qanon Republicans non-Qanon Republicans

3 / 8

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SLIDE 11

Visualizing QAnon support

+

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers Qanon Republicans non-Qanon Republicans

3 / 8

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SLIDE 12

Visualizing QAnon support

+

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers Qanon Republicans non-Qanon Republicans ℙ(혘)

3 / 8

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SLIDE 13

Visualizing QAnon support

+

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers non-Qanon Republicans ℙ(혙∣혘) ℙ(혘)

3 / 8

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SLIDE 14

Visualizing QAnon support

ℙ(혙∣혘)ℙ(혘) + ℙ(혙∣혘)ℙ(혘)

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers non-Qanon Republicans ℙ(혙∣혘) ℙ(혘)

3 / 8

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SLIDE 15

Visualizing QAnon support

ℙ(혙∣혘)ℙ(혘) + ℙ(혙∣혘)ℙ(혘)

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers non-Qanon Republicans ℙ(혙∣혘) ℙ(not 혘) ℙ(혘)

3 / 8

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SLIDE 16

Visualizing QAnon support

ℙ(혙∣혘)ℙ(혘) + ℙ(혙∣혘)ℙ(혘)

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers ℙ(혙∣혘) ℙ(혙∣not 혘) ℙ(not 혘) ℙ(혘)

3 / 8

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SLIDE 17

Visualizing QAnon support

ℙ(혙∣not 혘)ℙ(not 혘) ℙ(혙∣혘)ℙ(혘) + ℙ(혙∣혘)ℙ(혘)

Chance a random Republican believes QAnon =

Qanon nonbelievers Qanon believers ℙ(혙∣혘) ℙ(혙∣not 혘) ℙ(not 혘) ℙ(혘)

3 / 8

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SLIDE 18

Bayes’ rule

  • Reverend Thomas Bayes (1701–61): English minister and statistician
  • Bayes’ rule: if ℙ(𝘊) > 𝟣, then:

ℙ(𝘉 ∣ 𝘊) = ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) ℙ(𝘊) = ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) + ℙ(𝘊 ∣ not 𝘉)ℙ(not 𝘉)

4 / 8

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SLIDE 19

Bayes’ rule

  • Reverend Thomas Bayes (1701–61): English minister and statistician
  • Bayes’ rule: if ℙ(𝘊) > 𝟣, then:

ℙ(𝘉 ∣ 𝘊) = ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) ℙ(𝘊) = ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) + ℙ(𝘊 ∣ not 𝘉)ℙ(not 𝘉)

4 / 8

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SLIDE 20

Bayes’ rule

  • Reverend Thomas Bayes (1701–61): English minister and statistician
  • Bayes’ rule: if ℙ(𝘊) > 𝟣, then:

ℙ(𝘉 ∣ 𝘊) = ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) ℙ(𝘊) = ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) ℙ(𝘊 ∣ 𝘉)ℙ(𝘉) + ℙ(𝘊 ∣ not 𝘉)ℙ(not 𝘉)

4 / 8

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SLIDE 21

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

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SLIDE 22

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

slide-23
SLIDE 23

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

slide-24
SLIDE 24

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

slide-25
SLIDE 25

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

slide-26
SLIDE 26

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

slide-27
SLIDE 27

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

slide-28
SLIDE 28

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon)

ℙ(QAnon ∣ Republican)

  • How does the evidence change the chance of the hypothesis being true?

5 / 8

slide-29
SLIDE 29

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon) ⇝ ℙ(QAnon ∣ Republican)
  • How does the evidence change the chance of the hypothesis being true?

5 / 8

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SLIDE 30

Why is Bayes’ rule useful?

  • What is the probability of some hypothesis given some evidence?
  • ℙ(QAnon ∣ Republican)?
  • Often easier to know probability of evidence given hypothesis.
  • ℙ(Republican ∣ QAnon)
  • Combine this with the prior probability of the hypothesis.
  • Prior: ℙ(QAnon)
  • Posterior: ℙ(QAnon ∣ Republican)
  • Applying Bayes’ rule is often called updating the prior.
  • ℙ(QAnon) ⇝ ℙ(QAnon ∣ Republican)
  • How does the evidence change the chance of the hypothesis being true?

5 / 8

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SLIDE 31

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

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SLIDE 32

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

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SLIDE 33

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

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SLIDE 34

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

slide-35
SLIDE 35

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

slide-36
SLIDE 36

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

slide-37
SLIDE 37

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

slide-38
SLIDE 38

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

slide-39
SLIDE 39

Uses of Bayes’ rule

  • Medical testing:
  • Want to know: ℙ(Disease ∣ Test Positive)
  • Have: ℙ(Test Positive ∣ Disease) and ℙ(Disease)
  • Predicting traits from names:
  • Want to know: ℙ(African American ∣ Last Name)
  • Have: ℙ(Last Name ∣ African American) and ℙ(African American)
  • Spam fjltering:
  • Want to know: ℙ(Spam ∣ Email text)
  • Have: ℙ(Email text ∣ Spam) and ℙ(Spam)

6 / 8

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SLIDE 40

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-41
SLIDE 41

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-42
SLIDE 42

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-43
SLIDE 43

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-44
SLIDE 44

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-45
SLIDE 45

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-46
SLIDE 46

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-47
SLIDE 47

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-48
SLIDE 48

Medical tests

  • Suppose you go and get a COVID-19 test and it comes back positive!
  • Let a positive test be 𝘘𝘜 .
  • What’s the probability you actually have COVID-19?
  • Let having COVID be labeled 𝘋.
  • Question: What is ℙ(𝘋 ∣ 𝘘𝘜)?
  • Components for calculating Bayes’ rule:
  • ℙ(𝘘𝘜|𝘋) = 𝟣.𝟪: true positive rate
  • ℙ(𝘘𝘜 ∣ not 𝘋) = 𝟣.𝟣𝟨: false positive rate
  • ℙ(𝘋) = 𝟣.𝟣𝟣𝟤 rough prevalance of active COVID cases.

7 / 8

slide-49
SLIDE 49

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-50
SLIDE 50

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-51
SLIDE 51

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-52
SLIDE 52

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-53
SLIDE 53

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-54
SLIDE 54

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-55
SLIDE 55

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-56
SLIDE 56

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-57
SLIDE 57

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8

slide-58
SLIDE 58

Applying Bayes’ rule to COVID tests

  • Use the law of total probability to get the denominator:

ℙ(𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) + ℙ(𝘘𝘜|not 𝘋)ℙ(not 𝘋) = (𝟣.𝟪 × 𝟣.𝟣𝟣𝟤) + (𝟣.𝟣𝟨 × 𝟣.𝟬𝟬𝟬) = 𝟣.𝟣𝟨𝟤

  • Now plug in all values to Bayes’ rule:

ℙ(𝘋 ∣ 𝘘𝘜) = ℙ(𝘘𝘜 ∣ 𝘋)ℙ(𝘋) ℙ(𝘘𝘜) = 𝟣.𝟪 × 𝟣.𝟣𝟣𝟤 𝟣.𝟣𝟨𝟤 ≈ 𝟣.𝟣𝟤𝟧

8 / 8