From: Dan Fabulich (dfabulich@warpmail.net)
Date: Tue Jul 08 2003 - 22:35:37 MDT
Eliezer S. Yudkowsky wrote:
> >>>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?
> >
> > In this stupid problem, the authors badly misstate at least one
> > of the premises, in my opinion. If you draw a picture of the
> > problem, perhaps, you may be less likely to misread it than if
> > you plug the numbers into a formula.
>
> 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
Now, wait a minute.
You begin with p(deformity) = 0.01. I presume that's your interpretation
of the claim that "a newborn will have deformities traceable to a sickness
of its mother during pregnancy", but that doesn't seem right to me at all.
[Obviously p(deformity) should include cases where there are deformities
that aren't traceable to a sickness of the mother.]
Here's my interpretation of the problem.
p(deformity&traceable) = 0.01
p(rubella|~deformity) = 0.1
p(rubella|deformity&traceable) = 0.5
Where the question is: what's p(deformity|rubella)?
Now, perhaps I can assume that deformity&rubella entails traceable, but I
can't just assume that deformity->traceable. (And let's not even get into
the possibility that the child may not be "born healthy and normal" but
may have no "deformities".)
But unless we know p(deformity&~traceable), an obviously non-zero number,
we can't get p(deformity) or p(~deformity) at all, which we need to get
p(rubella&~deformity).
Of course, in light of what they think the answer is, it's obvious that
they think trivial details like these are irrelevant to the problem;
perhaps they think my point is a nitpick or something.
But I'm on Lee's side here... this problem is very badly stated.
> I would (as stated more briefly in my first reply) recommend that you read
> through the "Intuitive Explanation" from start to finish without skipping
> anything; 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.
I note here that he has worked out a number of very clear frequentist
accounts; accounts which don't suffer from as much vagueness as my own,
but which suffer from their own kinds of problems.
Note especially section 3.4 and Lecture 5 though his objections against
frequentism are available througout.
-Dan
-unless you love someone-
-nothing else makes any sense-
e.e. cummings
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