# Re: Fun With Bayes' Theorem

From: Eliezer S. Yudkowsky (sentience@pobox.com)
Date: Sun May 13 2001 - 21:12:03 MDT

Lee Corbin wrote:
>
> Eliezer Yudkowsky wrote:
>
> > Out of all the parents who *could* say "At least one of my children is a
> > girl", 2/3 of them have a boy. Out of all the parents who *do* say "At
> > least one of my children is a girl", 1/2 of them have a boy.
>
> I am sufficiently innocent of Bayesian statistics that I have to
> admit that anything is possible; if you're right, then there is
> a demarcation between what mathematicians would or should say,
> and what statisticians could or do say. :-)

(Vorlon voice:) "We are all Bayesian statisticians."

> But why are the parents who *do* say "At least..." different than
> the parents who *could* say it?

For the same reason that the set of people who can jump off buildings is
different from the set of people who do jump off buildings. I deny that
the distinction between "mathematician" and "statistician" is the right
one to make here. The mathematician is not being confronted with the fact
that a person *could* say something but the fact that the person *did* say
something. Now, if the mathematician were to ask the father "Is at least
one of your children a girl?" and the father answered "Yes", that would be
an *entirely different* matter. In fact, I bet that if you presented the
riddle this way, a much larger percentage of the reasoners would get it
right the first time.

Hey, I just made a testable prediction! Anyone want to try it? Robin?

> The earlier idea, who *could* truthfully
> say what, seems much easier to grasp. As for what people do say, this
> would seem to me to depend a great deal upon the particular situation.
> One easily contrives a situation where even under *do* I'm right: King
> Harod orders everybody who could say it to go to a mathematician and
> actually say it, etc.

Yes, that's why I said that the riddle does not adequately specify the
priors.

> What situations do you have in mind where
>
> > Out of all the parents who *do* say "At least one of
> > my children is a girl", 1/2 of them have a boy.

Situations where those parents who have both a boy and a girl will choose
at random whether to say "At least one of my children is a boy" or "at
least one of my children is a girl".

-- -- -- -- --
Eliezer S. Yudkowsky http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence

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