Re: Hempel's paradox redux

From: Eliezer S. Yudkowsky (
Date: Thu Sep 15 2005 - 16:48:59 MDT

Ben Goertzel wrote:
> FYI,
> Just to put the raven paradox to rest, I did a little poking around and
> found a paper
> which does a quite nice job of addressing all the issues that intuitively
> bothered me with the traditionally-cited Bayesian analyses of the Raven
> paradox.
> He shows why the traditional Bayesian arguments depend on unacceptable
> assumptions, but then gives a pretty rigorous mathematical analysis based on
> fairly minimal assumptions -- it's a nice paper. [In other words, he
> criticizes probability theory "as traditionally deployed" in this context,
> but does it well ;-) ] *This* kind of probabilistic argumentation is always
> convincing to me...

Actually, it looks like Fitelson is saying pretty much the same thing I
said in my first reply: Say explicitly your background assumptions
about sampling and prior probabilities, because the results will
strongly depend on them. Fitelson also analyzes the dependency on
background assumptions in essentially the same way - albeit at greater
depth and lesser breadth.

The main part of Fitelson's paper focuses narrowly on one challenge:
weakening the assumptions necessary to show that a non-black non-raven
is *much weaker* evidence favoring "All ravens are black" than is a
black raven, assuming both evidences are favorable. I read the paper
quickly, but it looks to me like I discussed a wider range of conditions
and dependencies in my own, fast, nonrigorous presentation. That is,
the main part of Fitelson's paper is a rigorous, narrow analysis of one
of many issues I briefly mentioned.

Where Fitelson and I discuss similar relations between background
assumptions and results, Fitelson's conclusions seem to agree with mine.
  Thus I don't see any place where I departed from the practice of good
Bayesians in my first reply to your question - I elaborated on the
traditional Bayesian analysis along the same lines Fitelson does. And
you did criticize my own reply to you, not only probability theory "as
traditionally deployed". Ben, I don't think you have a strong grasp on
Bayesian probability theory "as traditionally deployed" - at the very
least you need more practice applying it to example problems before you
develop a decent feel for it.

I also don't think it's fair to characterize Fitelson's paper as a
criticism arguing that "traditional" Bayesian analysis relies on hidden,
unacceptable assumptions. There are many published works on Hempel's
paradox by authors trying to work within the Bayesian tradition. Those
that I have seen tend to do a decent job of stating their assumptions,
albeit not with the rigor shown by the best Bayesians such as Jaynes.
Fitelson is not criticizing those analyses, he is strengthening a
certain result by weakening its assumptions. The part where Fitelson
tries to characterize the standard assumption as overly strong, is not
so much criticism of traditionally deployed Bayesian probability theory,
as Fitelson trying to show that his new result is interesting.

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

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