[extropy-chat] what is probability?

[gts]

On Tue, 16 Jan 2007 13:25:40 -0500, Benjamin Goertzel <ben at goertzel.org>  
wrote:

> I am not sure this observation about randomness is tied to
> frequentism, actually.

You may be right... I took a quick peek at those papers. I think I could  
spend at least several days just trying to understand what they are going  
on about. :) They appear to be about some concepts in information theory,  
mainly, not about the more modest and understandable (at least to me)  
undertaking that is probability theory.

For our purposes I think it's fair to say a sequence is random if there is  
and can be no discernible pattern, i.e., the sequence is random if the  
observations are *independent trials* in the usual sense.

> How would you define randomness of a finite entity **objectively**
> (independently of the observer) from a Bayesian point of view?

If I understand your question, you are really wanting know how randomness  
is defined subjectively. Yes? Otherwise I don't know how to make sense of  
your question. I say this because although there exists an animal called  
objective bayesianism, it is still an epistemic theory of probability.

As far as I know (and I could be wrong here) all bayesian views on  
randomness make use of a principle De Finetti called 'exchangeability'.  
'Exchangeability' is the subjectivist correlate to the objectivist idea of  
'independence'.

Very roughly speaking, exchangeability is true when you view any subset of  
subjective observations from a larger set as exchangeable in the equations  
with any other subset, with no consequence. (That's probably a terrible  
summation, but it's the best I can come up with at the moment. [1]).  
Exchangeable events are to subjectivists what independent/random events  
are to objectivists.

Interesting to me is the fact that on the subjectivist (bayesian) view,  
events are never independent! Even perfectly idealized random coin-flips  
are *not* considered 'independent trials'. The concept of independence has  
almost no use in the subjective view.

As de Finneti put it:

"If I admit the possibility of modifying my probability judgment in  
response to observation of frequencies; it means that - by definition - my  
judgment of the probability of one trial is not independent of the  
outcomes of the others."

This is also a weakness of the interpretation in my opinion. To my mind it  
is a bit absurd to think that coin-flips are not independent of one  
another.

> This is an interesting topic... ;-)

I think so too. :)

-gts

1. Here is a more technical explanation of exchangeability:
http://www-stat.stanford.edu/~cgates/PERSI/courses/stat_121/lectures/exch/