[extropy-chat] what is probability?

gts gts_2000 at yahoo.com
Tue Jan 16 20:40:52 UTC 2007

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

> 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  

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  

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

I think so too. :)


1. Here is a more technical explanation of exchangeability:

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