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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

  1. Gov 51: Bayes Rule Matthew Blackwell Harvard University 1 / 8

  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

  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

  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

  5. Visualizing QAnon support 3 / 8 Qanon nonbelievers

  6. Visualizing QAnon support 3 / 8 Qanon Qanon believers nonbelievers

  7. Visualizing QAnon support 3 / 8 Qanon Qanon believers nonbelievers Qanon Republicans

  8. Visualizing QAnon support 3 / 8 Qanon Qanon believers nonbelievers Qanon non-Qanon Republicans Republicans

  9. 3 / 8 Visualizing QAnon support Qanon Qanon believers nonbelievers Qanon non-Qanon Republicans Republicans Chance a random Republican believes QAnon =

  10. 3 / 8 Visualizing QAnon support Qanon Qanon believers nonbelievers Qanon non-Qanon Republicans Republicans Chance a random Republican believes QAnon =

  11. Visualizing QAnon support 3 / 8 Qanon Qanon believers nonbelievers Qanon non-Qanon Republicans Republicans Chance a random Republican believes QAnon = +

  12. 3 / 8 Visualizing QAnon support Qanon Qanon believers nonbelievers Qanon non-Qanon Republicans Republicans ℙ(혘) Chance a random Republican believes QAnon = +

  13. Visualizing QAnon support 3 / 8 Qanon Qanon believers nonbelievers ℙ(혙∣혘) non-Qanon Republicans ℙ(혘) Chance a random Republican believes QAnon = +

  14. Visualizing QAnon support 3 / 8 Qanon Qanon believers nonbelievers ℙ(혙∣혘) non-Qanon Republicans ℙ(혘) ℙ(혙∣혘)ℙ(혘) Chance a random Republican believes QAnon = + ℙ(혙∣혘)ℙ(혘)

  15. 3 / 8 Visualizing QAnon support Qanon Qanon believers nonbelievers ℙ(혙∣혘) non-Qanon Republicans ℙ( not 혘) ℙ(혘) ℙ(혙∣혘)ℙ(혘) Chance a random Republican believes QAnon = + ℙ(혙∣혘)ℙ(혘)

  16. Visualizing QAnon support 3 / 8 Qanon Qanon believers nonbelievers ℙ(혙∣혘) ℙ(혙∣ not 혘) ℙ( not 혘) ℙ(혘) ℙ(혙∣혘)ℙ(혘) Chance a random Republican believes QAnon = + ℙ(혙∣혘)ℙ(혘)

  17. 3 / 8 Visualizing QAnon support Qanon Qanon believers nonbelievers ℙ(혙∣혘) ℙ(혙∣ not 혘) ℙ( not 혘) ℙ(혘) ℙ(혙∣혘)ℙ(혘) Chance a random Republican believes QAnon = ℙ(혙∣ not 혘)ℙ( not 혘) + ℙ(혙∣혘)ℙ(혘)

  18. ℙ( 𝘉 ∣ 𝘊 ) = ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 ) Bayes’ rule • Reverend Thomas Bayes (1701–61): English minister and statistician • Bayes’ rule : if ℙ( 𝘊 ) > 𝟣 , then: ℙ( 𝘊 ) = ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 ) ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 ) + ℙ( 𝘊 ∣ not 𝘉 )ℙ( not 𝘉 ) 4 / 8

  19. Bayes’ rule • Reverend Thomas Bayes (1701–61): English minister and statistician • Bayes’ rule : if ℙ( 𝘊 ) > 𝟣 , then: ℙ( 𝘊 ) = ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 ) ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 ) + ℙ( 𝘊 ∣ not 𝘉 )ℙ( not 𝘉 ) 4 / 8 ℙ( 𝘉 ∣ 𝘊 ) = ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 )

  20. Bayes’ rule • Reverend Thomas Bayes (1701–61): English minister and statistician • Bayes’ rule : if ℙ( 𝘊 ) > 𝟣 , then: ℙ( 𝘊 ) = ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 ) ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 ) + ℙ( 𝘊 ∣ not 𝘉 )ℙ( not 𝘉 ) 4 / 8 ℙ( 𝘉 ∣ 𝘊 ) = ℙ( 𝘊 ∣ 𝘉 )ℙ( 𝘉 )

  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

  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

  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

  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

  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

  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

  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

  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

  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 . • How does the evidence change the chance of the hypothesis being true? 5 / 8 • ℙ( QAnon ) ⇝ ℙ( QAnon ∣ Republican )

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