Re: Fun With Bayes' Theorem

From: Eliezer S. Yudkowsky (
Date: Tue May 15 2001 - 20:25:26 MDT

Lee Corbin wrote:
> 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.

You can sub in "illustrating previously unknown reasoning techniques" for
"gloating" if you so prefer. My point is that someone who answers "I bet
you think it's 1/3, but it's actually 1/2" is demonstrating deeper
knowledge than someone who answers "1/3", just as someone who answers
"1/3" probably has more (declarative!) knowledge than someone who guesses

> 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.

There are non-Bayesians? I've never heard of any non-Bayesians or any
"controversy". Where I come from, "Bayesian reasoning" is practically a
synonym for "normative rationality".

Maybe you're right. Maybe a *mathematician* answers 1/3 - the problem was
posed about mathematicians, after all. And I bet that a statistician
answers 1/3 as well. However, a sufficiently alert cognitive scientist
should answer 1/2.

So now we have *four* possible answers:

"1/2." (naive human)

"1/3." (naive mathematician)

"You'd think it'd be 1/3, but it's actually 1/2." (bayesian)

"It's actually 1/2, but all but the most alert mathematicians would answer
1/3." (actual answer)

> Only one question remains:
> > (Vorlon voice:) "We are all Bayesian statisticians."
> 'zat so? You mean I missed out on the controversy after all?

No - I just mean that the human brain runs on folk bayesian reasoning
rather than folk mathematics. No, scratch that, that's a lousy sentence.
What I mean is that, in this instance at least, the Bayesian answer is
much closer to the way our brains work than the mathematical answer.

-- -- -- -- --
Eliezer S. Yudkowsky
Research Fellow, Singularity Institute for Artificial Intelligence

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