Bayes Theorem Bayess theorem Pr [ A | B ] = Pr [ B | A ] Pr [ A ] - PowerPoint PPT Presentation

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Bayes Theorem Bayess theorem Pr [ A | B ] = Pr [ B | A ] Pr [ A ] Pr [ B ] Note: This is very useful in both this course and in life. Example of Application of Bayess theorem Pr [ A | B ] = Pr [ B | A ] Pr [ A ] Pr [ B ] . There

Bayes Theorem
Bayes’s theorem ◮ Pr [ A | B ] = Pr [ B | A ] · Pr [ A ] Pr [ B ] Note: This is very useful in both this course and in life.
Example of Application of Bayes’s theorem Pr [ A | B ] = Pr [ B | A ] · Pr [ A ] Pr [ B ] . There are two coins: 1) Coin F is fair: Pr ( H ) = Pr ( T ) = 1 2 . 2) Coin B is biased: Pr ( H ) = 3 4 , Pr ( T ) = 1 4 . Alice picks coin at random, flips 10 times, gets all H. Is the coin definitely biased?
Example of Application of Bayes’s theorem Pr [ A | B ] = Pr [ B | A ] · Pr [ A ] Pr [ B ] . There are two coins: 1) Coin F is fair: Pr ( H ) = Pr ( T ) = 1 2 . 2) Coin B is biased: Pr ( H ) = 3 4 , Pr ( T ) = 1 4 . Alice picks coin at random, flips 10 times, gets all H. Is the coin definitely biased? No.
Example of Application of Bayes’s theorem Pr [ A | B ] = Pr [ B | A ] · Pr [ A ] Pr [ B ] . There are two coins: 1) Coin F is fair: Pr ( H ) = Pr ( T ) = 1 2 . 2) Coin B is biased: Pr ( H ) = 3 4 , Pr ( T ) = 1 4 . Alice picks coin at random, flips 10 times, gets all H. Is the coin definitely biased? No. What is Prob that it is biased? VOTE: 1. Between 0.99 and 1.0 2. Between 0.98 and 0.99 3. Between 0.97 and 0.98 4. Less than 0.97
Example of Application of Bayes’s theorem Pr [ A | B ] = Pr [ B | A ] · Pr [ A ] Pr [ B ] . There are two coins: 1) Coin F is fair: Pr ( H ) = Pr ( T ) = 1 2 . 2) Coin B is biased: Pr ( H ) = 3 4 , Pr ( T ) = 1 4 . Alice picks coin at random, flips 10 times, gets all H. Is the coin definitely biased? No. What is Prob that it is biased? VOTE: 1. Between 0.99 and 1.0 2. Between 0.98 and 0.99 3. Between 0.97 and 0.98 4. Less than 0.97 We will see that it is 0.982954, so between 0.98 and 0.99.
Example of Application of Bayes’s theorem Pr ( B | H 10 ) = Pr ( B ) Pr ( H 10 | B ) P ( H 10 ) Pr ( B ) = 1 2 Pr ( H 10 | B ) = ( 3 4 ) 10 Pr ( H 10 ) = Pr ( H 10 ∩ F ) + Pr ( H 10 ∩ B ) Pr ( H 10 ∩ F ) = Pr ( H 10 | F ) Pr ( F ) + Pr ( H 10 | B ) Pr ( B ) = � � 2 ) 10 + ( 3 1 ( 1 4 ) 10 2 Put it together to get 1 Pr ( B | H 10 ) = 1 + (2 / 3) 10 = 0 . 982954 .
Example of Application of Bayes’s theorem Pr ( B | H 10 ) = Pr ( B ) Pr ( H 10 | B ) P ( H 10 ) Pr ( B ) = 1 2 Pr ( H 10 | B ) = ( 3 4 ) 10 Pr ( H 10 ) = Pr ( H 10 ∩ F ) + Pr ( H 10 ∩ B ) Pr ( H 10 ∩ F ) = Pr ( H 10 | F ) Pr ( F ) + Pr ( H 10 | B ) Pr ( B ) = � � 2 ) 10 + ( 3 1 ( 1 4 ) 10 2 Put it together to get 1 Pr ( B | H 10 ) = 1 + (2 / 3) 10 = 0 . 982954 . 1 Pr ( B | H n ) = 1 + (2 / 3) n .

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