# Re: Fun With Bayes' Theorem

From: Lee Corbin (lcorbin@ricochet.net)
Date: Tue May 15 2001 - 19:55:30 MDT

Eliezer wrote
>Lee Corbin wrote:
>>
>> Choose at random? Here, I suspect, is where Bayesianism comes in.
>> The B statisticians are happy to assume certain priors when there
>> is no evidence for them. Now, I've always liked that, and feel
>>
>> But in some cases, like this, they start reading into situations
>> protocols that aren't needed to answer the question.
>
>Which is all fine and good except when it gives you the WRONG ANSWER.

This is begging the question, of course.

>In this case, "could" may be more convenient (why?) but it is
>still the wrong way to define the mathematical problem given
>the word problem.

It's very much more convenient not to have to specify hypotheses.
But more importantly, from the statement "I have two children and
one is a girl", we don't (in mathematics) have to think about
whether or not King Herod has been ordering parents of girls
to appear somewhere and give evidence, or whether a parent flips
his children into the air and reports on the one that lands first,
or whether some entirely other real-world procedure was employed.
Mathematics is the art of abstracting away all of that. Moreover,

the art of mathematical problems consists in finding the relevant
pure abstract situation that applies.

>Our entire real world consists of "did", not "could". "Could"
>is the rational construct; "did" is the observed reality.

Well, not quite. Should we go further and find out an empirical
answer to the question "What did the mathematician say?"? After
all, (talking about the real world), how many people really *do*
take such a statement to a mathematician? Should we go test
whether the random processes employed were reliable? (Real
statisticians would!) The whole situation of my problem is
clearly contrived, and even meant to be funny (like what kind
of idiot would take his doubts about the sex of his unborn
child to a mathematician)?

In the real world, statisticians get real problems from real
customers. It is expected that they will thoroughly research
any number of hypotheses, often taking weighted averages of
what they find, in order to submit the most probably guess
about a real world phenomenon. This is admirable, of course,
but it is entirely different from the world of asking little
mathematics puzzles.

>If you want to pose a mathematical problem, you should use a
>pure mathematical language, or a word problem with no flexibility
>in the priors.

Again, whether or not the priors are flexible is a question that
I've *never* seen in mathematics problems. This is definitely
a statisticians approach. Ask mathematicians "what are the priors?"
and in many cases you'll get a blank look; only those, I submit,
who are acquainted with statistical procedures will instantly know
what you are talking about.

>To put it another way, Bayesian probabilities automatically
>govern all problems that are posed in real-world language, and

Right. Hopefully no one would think "a man, uncertain what the
sex his unborn child was, decided to consult a mathematician" is
a real world problem.

>...if you want to pose a purely mathematical question in that
>language, you need to eliminate any Bayesian ambiguities. If
>you don't, then somebody who gives a "common-sense but wrong
>answer" that is actually correct gets to gloat over you
>instead of vice versa.

I'm sorry; you're quite right. *Gloating* is never justified,
and the real purpose of cute mathematical problems is the same
as the purpose of any interesting tidbit that one might want
to throw out. We're not trying to just show off; we expose our
beliefs to criticism, and also hope that there will be an
exchange of wisdom. We hope to be enlightened.

Well, this has been enlightening indeed! All my life I felt
deprived because I didn't know what the controversy surrounding
the "Bayesians" and "Non-Bayesians" was, and at one point, it
was the only intellectual controversy I knew of that I felt
excluded from. Perhaps I've finally gotten involved.

Even better, my problem about the man who didn't know the sex
of his unborn child---and I elided a sentence about how he wanted
a boy, and so "cleverly" went to a mathematician---now evidently
has even richer structure: it can be used to explain the difference
between a Bayesian and a Non-Bayesian approach. Only one question
remains:

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

'zat so? You mean I missed out on the controversy after all? :-(

Lee

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