[extropy-chat] Paradox? What paradox?
gts
gts_2000 at yahoo.com
Sat Jan 6 15:43:12 UTC 2007
On Sat, 06 Jan 2007 00:52:03 -0500, Rafal Smigrodzki
<rafal.smigrodzki at gmail.com> wrote:
> I'd rather say the problem is a trick question rather than a paradox :)
It's a real paradox, in the class of paradoxes known generically as
Bertrand' Paradox(es). However they are paradoxes iff we assume the
principle of indifference.
Here is a different version of the same paradox:
http://www.cut-the-knot.org/bertrand.shtml
(This is the version of Bertrand's Paradox to which Jaynes offered a
possible resolution. But as I mentioned to Ben, it's my understanding that
Jaynes' apparent solution does not apply to every version of Bertrand's
Paradox and so cannot be considered a real solution.)
> The problem may seem vexing at first glance, until one notices that the
> procedure used by the factory to choose which cube to make is not
> defined in the
> formulation of the problem.
I might say the procedure *is* defined in the problem; that it is defined
as a "random" procedure.
> therefore the event of "randomly choosing" a cube is not defined
> sufficiently
Yes. The paradox goes away, as I think you see, when we realize we have no
idea what we mean we speak of selecting a "random" cube or of performing a
"random selection procedure".
More generally, the principle of indifference seems not to apply to
continuous variables. (I mentioned this paradox in the first place by way
of criticising the principle of indifference upon which some epistemic
theories of probability depend.)
We can say "with respect to a random coin-flip, if we have no evidence to
favor either heads or tails then we should assume the two possible
outcomes to have equal probabilities". We are indifferent to the two
outcomes and so the two outcomes receive the same subjective probability.
This is the (supposed) principle of indifference in action.
However we run into trouble when we say things like "with respect to a
random cube selection (as per the paradox) we should assume the two
possible outcomes have equal probabilities." Here again we are indifferent
to the two possible outcomes, just as in the coin-flip example above, but
the principle of indifference fails miserably.
-gts
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