# RE: Hempel's paradox redux

From: Ben Goertzel (ben@goertzel.org)
Date: Thu Sep 15 2005 - 06:38:13 MDT

Hey Eli,

> I find your lack of faith disturbing.
>
> I've got to sleep now, so this is a fine opportunity for aspiring
> probability theorists on SL4 to test their wings. Props for anyone who
> points out flaws in Ben's reasoning before I write a response tomorrow.

Well, my wife and I found the flaw in a brief conversation before we went to
sleep last night but I was really tired and I didn't feel like dealing with
the computer anymore ;)

As she pointed out, the situation with states 2 and 4 in my little story
about the midget is a lot like the situation where you have two coins, one
with two heads and one regular coin with a head and a tail. If you are told
that one has been tossed and the result was a head, then the odds are 2:1
that the the coin in question was the double-headed coin. [The heads are
W's, the tail is a B]

Duuuuhhh...

> These props are also available to Ben, but if he doesn't get it on his
> own, and I have to write up the flaws myself, I want Ben to never again
> claim that Bayesian probability is quote wrong unquote.

I wouldn't every say something like "probability theory is wrong" -- it's a
branch of math and is correct assuming its axiom systems, just like any
other branch of math.... It's even a *very useful* branch of math which is
why Novamente is substantially based upon it....

If you show me a particular theorem of probability theory I will always
agree it's right, assuming it's not above my algorithmic information to
verify it ;-)

Issues of how to use probability theory to model some particular situation,
or whether it is the best tool to model some particular situation, are
different.... A way of using probability theory can be wrong even though
probability theory itself is not wrong.

In this particular case (the Hempel paradox), the standard way of using
probability theory seems to be right ... but that doesn't mean everyone's
way of using probability theory will always be right....

The real question about the Hempel paradox is not whether the probabilistic
analysis is mathematically correct, but whether the probabilistic notion of
"evidence" is the best one for formalizing the intuitive notion of evidence
underlying the Hempel paradox. I had thought for a moment the answer was
"no", but upon observing that my PTL notion of evidence (though formulated
differently) gives the same result as the standard probabilistic way of
thinking about evidence, I have had to reconsider. This is interesting for
me, from a more general perspective, because it is causing me to revisit
various other aspects of PTL and observe the various connections between
them and more conventional ways of formulating things probabilistically....
My objective in designing PTL was to create a convenient way for applying
probability theory to AGI, and I think/hope it's succeeded in that, but I am
seeing that the PTL quest didn't lead quite as far from conventional ways of
probability theory as I was thinking....

-- Ben

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