Tying together recent threads on indefinite probabilities and prior
distributions (PI, maxent, Occam)...
For those who might not know, the PI (the principle of indifference)
advises us, when confronted with n mutually exclusive and exhaustive
possibilities, to assign probabilities of 1/n to each of them.
In his book _The Algebra of Probable Inference_, R.T. Cox presents a
convincing disproof of the PI when n = 2. I'm confident his argument
applies for greater values of n, though of course the formalism would be
more complicated.
His argument is by reductio ad absurdum; Cox shows that the PI leads to an
absurdity. (Not just an absurdity in his view, but a "monstrous" absurdity
:-)
The following quote is verbatim from his book, except that in the interest
of clarity I have used the symbol "&" to mean "and" instead of the dot
used by Cox. The symbol "v" means "or" in the sense of "and/or".
Also there is an axiom used in the argument, referred to as "Eq. (2.8 I)".
That axiom is
(a v ~a) & b = b.
Cox writes, concerning two mutually exclusive and exhaustive propositions
a and b...
==========
...it is supposed that
a | a v ~a = 1/2
for arbitrary meanings of a.
In disproof of this supposition, let us consider the probability of the
conjunction a & b on each of the two hypotheses, a v ~a and b v ~b. We have
a v b | a v ~a = (a | a v ~a)[b | (a v ~a) & a]
By Eq (2.8 I) (a v ~a) & a = a and therefore
a & b | a v ~a = (a | a v ~a) (b | a)
Similarly
a & b | b v ~b = (b | b v ~b) (a | b)
But, also by Eq. (2.8 I), a v ~a and b v ~b are each equal to (a v ~a) &
(b v ~b) and each is therefore equal to the other.
Thus
a & b | b v ~b = a & b | a v ~a
and hence
(a | a v ~a) (b | a) = (b | b v ~b) (a | b)
If then a | a v ~a and b | b v ~b were each equal to 1/2, it would follow
that b | a = a | b for arbitrary meanings of and b.
This would be a monstrous conclusion, because b | a and a | b can have any
ratio from zero to infinity.
Instead of supposing that a | a v ~a = 1/2, we may more reasonably
conclude, when the hypothesis is the truism, that all probabilities are
entirely undefined except these of the truism itself and its
contradictory, the absurdity.
This conclusion agrees with common sense and might perhaps have been
reached without formal argument, because the knowledge of a probability,
though it is knowledge of a particular and limited kind, is still
knowledge, and it would be surprising if it could be derived from the
truism, which is the expression of complete ignorance, asserting nothing.
===========
-gts
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