1. An Intuitive Explanation of Bayes' Theorem By Eliezer Yudkowsky

2. reference  http://yudkowsky.net/rational/bayes http://yudkowsky.net/rational/bayes

3. Questions  100 out of 10,000 women at age forty who participate in routine screening have breast cancer. 80 of every 100 women with breast cancer will get a positive mammography. 950 out of 9,900 women without breast cancer will also get a positive mammography. If 10,000 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?

4. Compute  A= positive mammographies & actually have breast cancer  B = positive mammographies  results = A/B *100%  A=100*(80/100)=80  B= women with breast cancer with a positive mammography (A) + women without breast cancer with a positive mammography (C)  B= 100 * (80/100) +(10000-100) *(950/9900)  results = A/B *100% =80/1030=7.8%

5. Questions – different version  1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?  A=1%*80%  B=A+(1-1%)*9.6%  Results=A/B*100%=7.8%

6. Egg problem  Some eggs are painted red and some are painted blue. 40% of the eggs in the bin contain pearls, and 60% contain nothing. 30% of eggs containing pearls are painted blue, and 10% of eggs containing nothing are painted blue. What is the probability that a blue egg contains a pearl?  p(pearl) = 40%  p(blue|pearl) = 30%  p(blue|~pearl) = 10%  p(pearl|blue) = ?  "~" is shorthand for "not", so ~pearl reads "not pearl".  blue|pearl is shorthand for "blue given pearl“  "the probability that an egg is painted blue, given that the egg contains a pearl".  the order of implication is read right-to-left  blue|pearl means "blue<-pearl", the degree to which pearl-ness implies blue-ness  reads as "the probability that a particle at A goes to B, then to C, ending up at D".

7. Visualized Results

8. More notation  The item on the right side = what you already know or the premise,  The item on the left side = the implication or conclusion.  p(blue|pearl) = 30%,  we already know that some egg contains a pearl, then we can conclude there is a 30% chance that the egg is painted blue.  p(pearl|blue)  "the chance that a blue egg contains a pearl" or  "the probability that an egg contains a pearl, if we know the egg is painted blue"  p(pearl|blue) = p(pear&blue) / p(blue)

9. Bayes' Theorem  A=1%*80%  B=A+(1-1%)*9.6%  Results=A/B*100%=7.8%  A=p(cancer)*p(positive|cancer)  B=A+ p(~cancer) *p(positive|~cancer)  A/B= p(cancer|positive)

10. Bayes' Theorem  What we know  P(A)=15%  P(E| A =10%)  What we also know  P(~A)=1- 15% =85%  P(E|~ A) =80%  Why not 1- 10% ?  What we want to know  Probability of area ? in given E  P(A|E)=?  How? A E  p(E) ))p(AA|p(E E)|p(A  ))p(AA|p(E ))p(AA|p(E ))p(~A~A|p(E+

11. Bayes' Theorem A1A1 A2A2 A3A3 A4A4 A5A5 A6A6 E where {A i } forms a partition of the event space,partition Based on definition of conditional probability p(A i |E) is posterior probability given evidence E p(A i ) is the prior probability P(E|A i ) is the likelihood of the evidence given A i p(E) is the preposterior probability of the evidence   j jj iiii i ))p(AA|p(E ))p(AA|p(E p(E) ))p(AA|p(E E)|p(A

12. likelihood*prior Posterior= evidence  ii i p(E) ))p(AA|p(E E)|p(A

13. Example  You go to the doctor’s office, where you take a test for a horrible disease  The test is 99% accurate  If the test is positive, 99% of the time if you have the disease, negative 99% if you don’t  The disease itself is rare: occurs in I in 10,000 people  Your test is positive. What is the probability you have the disease?

14. 14 Why is this useful?  Useful for assessing diagnostic probability from causal probability  P(cause|effect)= P(effect|cause)P(cause) P(effect)  Let M be meningitis, S be stiff neck P(m|s)=P(s|m)P(m) = 0.8 X 0.0001 = 0.0008 P(s) 0.1  Note: posterior probability of meningitis is still very small!

15. Homework  Write a simulation for Monty Hall problem  Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?