From: Lee Corbin (lcorbin@tsoft.com)
Date: Wed Jul 09 2003 - 00:36:30 MDT
Eliezer writes
> >>>3. The probability that a newborn will have deformities
> >>> traceable to a sickness of its mother during pregnancy is 1%.
> >>> If a child is born healthy and normal, the probability that
> >>> the mother had rubella during her pregnancy is 10%. If a
> >>> child is born with deformities that can be traced to a
> >>> sickness of the mother, the probability that the mother had
> >>> rubella during her pregnancy is 50%. What is the probability
> >>> that a child will be born with deformities if its mother had
> >>> rubella during her pregnancy?
>
> No, Lee, it's perfectly straightforward to solve the problem from these
> premises.
>
> Observe:
>
> p(deformity) = 0.01
> p(~deformity) = 0.99
> p(rubella|~deformity) = 0.1
> p(rubella|deformity) = 0.5
> p(rubella&~deformity) = p(r|~d)p(~d) = .1*.99 = .099
> p(rubella&deformity) = p(r|d)p(d) = .5*.01 = .005
> p(rubella) = p(r&d) + p(r&~d) = .104
> p(deformity|rubella) = p(rubella&deformity)/p(rubella) = .005/.104 = .048
The problem is, as I think Dan has made you aware, that
you were assuming that "deformities" = "deformities traceable
to a sickness of the mother". By thinking immediately of the
odds ratio technique, you assumed that the problem was well-
formed and simple, and that necessarily would succumb to
just plugging in numbers in the usual way. This prevented
you from actually reading all the statements carefully.
Do you see which one is especially faulty here (at least in
your eyes)?
> > The probability that a newborn will have deformities
> > traceable to a sickness of its mother during pregnancy is 1%.
> >
> > If a child is born healthy and normal, the probability
> > that the mother had rubella during her pregnancy is 10%.
> >
> > If a child is born with deformities and it can be traced
> > to some sickness of the mother, the probability that the
> > mother had rubella during her pregnancy is 50%.
> >
> > What is the probability that a child will be born with
> > deformities if its mother had rubella during her pregnancy?
> ...
>
> It may not be a good idea to read academic papers about Bayes before
> having grasped Bayes,
I shall <sniff> demand a formal apology for this slur. 8^D
> If you look at the surrounding context, you will see that the problem they
> are addressing is that some incorrect answers computed according to common
> incorrect strategies are numerically close to the correct answer, and
> might be confused with a "Bayesian" answer by some common experimental
> techniques that accept answers as "Bayesian" if they are within a given
> range of the correct answer.
Oh, all right. It just looks as though they are saying p(HD) != P(H)*p(D|H):
The most frequent non-Bayesian algorithms they identified
include computing p (H&D) by multiplying p (H) and p (D | H);
because they are giving points to the test-takers only for the
use of Bayes formula---and, apparently, only to using the
posterior/prior odds formulation of it no less---(and in a
faulty problem to boot).
> Lee, Gigerenzer is extremely unlikely (prior probability) to be full of
> shit.
That's why it took me so long to figure out what was going on.
I couldn't believe that the problem---which I cut and pasted
for a class I was teaching---was bad. Fortunately, the kids
I was teaching worked enough on other Bayesian word problems
along this line, and I never inflicted this on them. (I nearly
did: "Hey, I can't figure this one out, but maybe you can.")
> and then, if you are still interested in understanding the
> difference between Bayesians and frequentists (which is something the
> intro does not address), I would recommend reading the E.T. Jaynes
> lectures given in the "Further Reading" section at the end of the intro.
> Jaynes gives specific examples of cases where frequentist methods are both
> more complicated than and inferior to Bayesian methods.
Yes, I've examined his book on-line, but then they took it offline
while the book was being printed. Last time I checked, it was back.
But the book was supposed to be published this year (all 2000 pages
or whatever of it), and that's what I want to get.
I do need better examples of where the Bayesians and non-Bayesians
disagree. (I have wanted to be in on that controversy since 1972,
and always felt side-lined.) I hope that not all Jaynes' examples
have to do with prior distributions, unbiased estimators,
transformations of likelihood, and so on.
Lee
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