[extropy-chat] what is probability?

[gts]

On Mon, 01 Jan 2007 13:54:42 -0500, Ben Goertzel <ben at goertzel.org> wrote:

> The most consistent interpretational approach, I believe, is a fusion
> of the Subjective Theory and Logical Theory as enabled by Cox's
> Theorem.

As promised I looked into this subject of Cox's Theorem. Thanks again for  
mentioning it.

As I understand now, and I hope you will correct me if I'm wrong, the  
"fusion of the Subjective Theory and Logical Theory" to which you refer is  
none other than what is more commonly known both as Objective Bayesianism  
and Logical Bayesianism (as distinct from Subjective Bayesianism). Yes?

If so then this is an interpretation about which I had already done some  
reading [1]. I had rejected it on the grounds that it depends on The  
Principle of Indifference, which leads to many unsolved and possibly  
unsolvable paradoxes. Was I wrong?

Here is what I mean by the Principle of Indifference leading to unsolved  
paradoxes:

First a definition of The Principle of Indifference:

"When there is no evidence favoring one possibility over another, the  
possibilities have the same probability."

For example if you have no knowledge about a coin's fairness or lack of  
fairness then it makes sense (seemingly) to assign equal probabilities to  
heads and to tails. You don't know if it's fair or not, and assuming it's  
not fair then you still don't know if it favors heads or if it favors  
tails. So we apply the principle of indifference and assign 50%  
probability to heads and 50% probability to tails. Logical, yes? So it  
would seem.

However this kind of thinking leads to paradoxes like the following  
(adapted from something I read on some forgotten web page)...

A factory produces cubes with side-length between 0 and 1 foot; what is  
the probability that a randomly chosen cube has side-length between 0 and  
1/2 a foot?

The tempting answer is 1/2, as we imagine a process of production that is  
uniformly distributed over side-length.

But the question could have been given an equivalent restatement:

A factory produces cubes with face-area between 0 and 1 square-feet; what  
is the probability that a randomly chosen cube has face-area between 0 and  
1/4 square-feet?

Now the tempting answer is 1/4, as we imagine a process of production that  
is uniformly distributed over face-area.

The problem could have been restated equivalently again:

A factory produces cubes with volume between 0 and 1 cubic feet; what is  
the probability that a randomly chosen cube has volume between 0 and 1/8  
cubic-feet? Now the tempting answer is 1/8, as we imagine a process of  
production that is uniformly distributed over volume.

And so on for all of the infinitely many equivalent reformulations of the  
problem (in terms of the fourth, fifth, … power of the length, and indeed  
in terms of every non-zero real-valued exponent of the length).

What, then, is the probability of the event in question???

---

So, although the principle of indifference seems "logical", I think it  
fails on closer inspection.

(I understand Jaynes tried with some apparent success to solve a related  
paradox, but from what I understand his solution has no universal  
application.)

-gts

1. Gillies, D. (2000). Philosophical Theories of Probability. New York,  
NY: Routledge.