Thinking Fast and Slow - Probability question

#1
In his book (Thinking Fast and Slow) Kahneman states: For example, if you believe that 3% of graduate students are enrolled in computer science (the base rate), and you also believe that the description of Tom W is 4 times more likely for a graduate student in that field than in other fields, then Bayes’s rule says you must believe that the probability that Tom W is a computer scientist is now 11%. If the base rate had been 80%, the new degree of belief would be 94.1%.

I would appreciate it if someone could show the math here. Not being able to do it is driving me crazy.
 
#3
Yes - it's a quote. What he is saying that out of a base of 100 students, 3 or 3% typically enroll in computer science. If Tom W is 4x as likely as that base rate, then the probability that he enrolls in Computer science is 11% - based on Baye's rule.
 

Jake

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#4
There's no other information? It looks to me like we don't have all the information that we need. .03 is clearly the base rate, P(H). The "4 times more likely" thing, as best I can tell, is the normalizing constant P(D). Converting from odds to probability we have P(D) = 4/(4+1) = .8. But to finish it we need the likelihood P(D|H), the probability of a description matching that of a computer scientist given that the person is in fact a computer scientist.
 
#5
If we know that, on average, out of a population of 100, 3 people will be computer scientists - then we have another person who is 4 times as likey as the average to be a computer scientist, can we say what the probability is that he will indeed be a computer scientist. That, I think is what he is saying - that the answer is a probability of 11% or 11 out of a hundred - based on Baye's rule.
 

Jake

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#6
Yes, I understand. I'm saying that that isn't enough information. Bayes' rule contains 4 terms, and we've only been given 2 pieces of information. So we have 1 equation with 2 unknowns. As stated, there are an infinity of solutions. Now, we could plug in the purported answer and solve for what the missing piece of information must be, but that seems to be missing the point.

Let's consider the problem in more concrete terms. We have a population of 1000 students, of which 30 are computer scientists (CS). All 1000 of these students have a personality sketch associated with them. Some of these sketches are CS-stereotypical and some are not CS-stereotypical. We believe that a sketch matching that of Tom W (that is, a CS-stereotypical sketch) is 4 times more likely for a graduate student in CS than in other fields. (Earlier I guessed that this piece of information was telling us about P(D), but on further reflection I think it is P(D|H)--like Dason mentioned, it is unclear.) So we believe that of these 30 computer science students, 4/(4+1) = 0.8 of them have a CS-stereotypical sketch, meaning 24 out of the 30 CS majors have a CS-stereotypical sketch. Now, we wish to know: given that Tom W has a CS-stereotypical sketch, what is the probability that he is a CS major? But we can't tell--it depends on how many of the 970 non-CS majors also happen to have a CS-stereotypical sketch (or, equivalently, how many students in the total population have a CS-stereotypical sketch)! And we haven't been told that.

For example, let's suppose that only 10% of the non-CS major students have a CS-stereotypical sketch (as opposed to the CS majors, 80% of whom do). This means that there are 97 non-CS majors who have CS-stereotypical sketches. So in the total population of students, we have 121 students who have a CS-stereotypical sketch (24 + 97), only 24 of whom are CS majors. So the probability that someone is a CS major given that they have a CS-stereotypical sketch in this case is 24/121 = .198. But now suppose instead that 50% of non-CS majors have a CS-stereotypical sketch. So now there are 24 + 485 = 590 total students with a CS-stereotypical sketch, and the probability that a student is a CS major given that they have a CS-stereotypical sketch is only 24/590 = .041.

The point is, the answer depends on how many CS-stereotypical sketches (like the sketch of Tom W) there are in the total population, which we haven't been told.
 

Jake

Cookie Scientist
#9
Actually, I thought about this yet some more and realized that the answer as given by Kahneman is correct. (I stand by my accusation however that the phrasing does not make the proper interpretation at all obvious!) I interpreted the "4 times as likely" thing as a simple odds, but it is actually an odds ratio; that is, it says not that the odds for a CS major to have a CS-stereotypical sketch are 4 to 1, but that the odds for a CS major are 4 times the odds for a non-CS major (whatever the actual odds are). So we do have all of the necessary information. Using the odds form of Bayes rule (LINK) we have that the likelihood ratio term is 4, and the base rate in odds is .03/(1+.03) = .029, giving us 4*.029 = .116, presumably the 11% figure that Kahneman quoted.
 

Jake

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#10
Hmm, this is a comedy of errors... 0.116 is the odds of being a CS major given the CS-stereotypical sketch, not the probability. The probability would be .116/(1-.116) = .131, which does not match Kahneman's answer. (Although if I plug in the second base rate that he gives, 0.8, I do get his second answer.) Someone check me on this?
 

Dason

Ambassador to the humans
#11
See I could get an answer that matches: .8*.03/(.8*.03 + .2*.97) but the interpretation of those quantities doesn't feel right.
 

Jake

Cookie Scientist
#12
Humbug... so I guess 4 times as likely is in terms of probabilities in this case? I tend to assume it refers to odds but the math doesn't work out that way. I feel like it shouldn't be this ambiguous...
 
#13
On another boatrd someone replied this (but I can't make it work with the 80% base rate):


Found this in the back of the book:

"For the hypothesis that Tom W is a computer scientist, the prior odds that correspond to a base rate of 3% are (.03/.97 = .031). Assuming a likelihood ratio of 4 (the description is 4 time as likely if Tom W is a computer scientist than if he is not), the posterior odds are 4 x .031 = 12.4. From these odds you can compute that the posterior probability of Tom W being a computer scientist is now 11% (because 12.4/112.4 = .11)."

By the way, I found this by googling:

Thinking Fast and Slow computer scientist is now 11%

and found it in Google Books.
 
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Jake

Cookie Scientist
#14
Okay, I see my persistent problem... I kept getting the odds-to-probability conversion and the probability-to-odds conversion exactly reversed! Those **** things are so similar I always get them mixed up.
 

Jake

Cookie Scientist
#16
Yep.

Convert new base rate to odds:
.8/(1-.8) = 4

Multiply by likelihood to get posterior odds:
4*4 = 16

Convert back to probability:
16/(1+16) = 0.941
 
#17
To my surprise (I'm not a math person) I used an Excel spreadsheet and got both sets of numbers to work:

.03 (prior probability) * 4 (conditional probability) = .12 .97 (prior probability) * 1 (conditional probability) = .97.

.12/(.12+.97) = 11


.08 * 4 = 3.2 .2 * 1 = .2

3.2/(3.2 + .2) = .941
 
#18
To my surprise (I'm not a math person) I used an Excel spreadsheet and got both sets of numbers to work:

.03 (prior probability) * 4 (conditional probability) = .12 .97 (prior probability) * 1 (conditional probability) = .97.

.12/(.12+.97) = 11


.08 * 4 = 3.2 .2 * 1 = .2

3.2/(3.2 + .2) = .941