[extropy-chat] what is probability?

[gts]

On Sun, 14 Jan 2007 21:36:50 -0500, Russell Wallace  
<russell.wallace at gmail.com> wrote:

> I'm inclined to think that issue would be best discussed in the context  
> of a practical example - do you have one to hand?

Sure.

An urn contains 100 green and/or red marbles in unknown proportion. We  
want to know pr(green) (the probability of drawing a green marble).

For better or worse, the principle of indifference (PI) tells us that  
given no evidence to prefer either green or red, we should assign them  
equal probabilities. So pr(green) = 0.5.

We set .5 as the prior probability in bayes' theorem, then begin  
performing experiments (drawing marbles with replacement), updating our  
judgmental probability along the way according to the theorem (bayesian  
conditioning).

After we've conditioned on enough evidence our judgmental probability  
should match, in the limit, the true objective chance of drawing a green  
marble from the urn (assuming we believe in such things as objective  
chance).

This is I think not a totally unreasonable process, even though our  
initial prior probability judgment was based on what you and I agree is  
shaky reasoning.

Opposing the PI is the idea that given no evidence, we should assign no  
probability at all, i.e., that we should accumulate some empirical  
evidence *before* we start estimating probabilities. But this idea makes  
some bayesians very uncomfortable.

-gts