[extropy-chat] what is probability?

[gts]

"The Ramsey-De Finetti theorem is a remarkable
achievement, and clearly demonstrates the superiority
of the subjective to the logical theory. Whereas in
the logical theory the axioms of probability could
only be justified by a vague and unsatisfactory appeal
to intuition, in the subjective theory they can be
proved rigorously from the eminently plausible
condition of coherence [...] In addition, the
subjective theory solves the paradoxes of the
Principle of Indifference by, in effect, making the
principle unnecessary, or at most a heuristic device."


-D. Gillies, (2000) _Philosophical Theories of
Probability_: Routledge

Concerning the paper cited by Ben in which someone
named Durack (if I have his name correctly) proposed a
solution to the wine/water paradox, I've been thinking
about that (always a bad sign)...

The author proposes a reasonable solution to the
wine/water paradox, but I've been thinking about the
general applicability of his idea to the bertrand
paradoxes.

The main idea in that paper is that paradoxes of the
Principle of Indifference (the PI) can be avoided if
we ask the right questions. Specifically the author
makes a persuasive argument that the probability of an
outcome is always conditional on the sample space that
contains it, and that the PI paradoxes go away if only
information about the relevant sample space is made
explicit in the questions that lead to the supposed
paradoxes. 

But something about that idea just seems too obvious..
and it occurs to me why: it is obvious because it's
nothing new at all, but rather just a sort of
restatement of the classical theory of Pascal, Fermat
and Laplace. 

The basic problem still remains in the case of, for
example, selecting the random cube. What is the
probability of the event in question? We're inclined,
after reading that paper, to reply, "The answer
depends on our choice of sample space, information not
included in the question." And of course there is
nothing at all wrong with our reply.

But here is the rub:

The principle of indifference exists as a supposed
logical principle for us to use in exactly this type
of situation of high uncertainty! To answer that "more
information is needed" is to admit that the PI has
failed us. 

Real-world situations can arise in which probability
decisions must be made with little or no knowledge of
the distribution or sample space, situations analogous
to that which we face in the aforementioned paradox of
selecting a random cube from the output of a cube
factory. It seems natural to invoke the principle of
indifference in such situations - it's certainly not
*incorrect* use it as a way to determine Bayesian
priors - but somehow I think it is nevertheless
incorrect to imagine that our decision to invoke the
PI is always strictly *rational*. 

So I'm finding myself back in agreement here with
Gillies above, that the PI is "at most a heuristic
device"...

-gts