[extropy-chat] what is probability?

[Benjamin Goertzel]

Hi,

The classic work on defining randomness is:

P. Martin-Lof " On the concept of a random sequence " , Theory
Probability Appl. 11: 177-179, (1966)

Roughly speaking: No computable function betting on the bits of a
Martin-Lof random sequence can make arbitrarily large amounts of
money.

This is known to be equivalent to Chaitin's definition of randomness
as incompressibility, see

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/ait9.html

and see

http://www.mcs.vuw.ac.nz/math/papers/martingale_final.pdf

for more recent work.

An extension to quantum computing (random qubit series) is here:

http://adsabs.harvard.edu/abs/1999chao.dyn..9031S

Note that all these definitions are about the randomness of infinitely
long sequences, hence not too useful for finite real-world situations.
 It seems that the notion of randomness is definable objectively only
for infinite entities.  To define randomness for finite entities one
needs to introduce an observer and define X as random if "observer O
can see no patterns in X."

How does this help with the PI and so forth?  Not at all as far as I
can tell, it's just the only work I know that formally addresses the
issue of defining randomness ;-)

-- Ben G



On 1/16/07, gts <gts_2000 at yahoo.com> wrote:
> On Mon, 15 Jan 2007 18:44:23 -0500, Rafal Smigrodzki
> <rafal.smigrodzki at gmail.com> wrote:
>
> > ### I would see it this way: The meaning of "random" is "obeying the
> > principle of indifference, where the sample space is unambiguously
> > described". If the sample space is exactly two outcomes, then each one
> > must occur 50% of the time, or else the coin is weighted, and the
> > tosses are not quite random anymore.
>
> Hmm, I think this is not at all what is or should be the definition of
> "random".
>
> As the word is normally defined, a series of tosses of an unfair coin
> still result in a completely random sequence! How can you suggest
> otherwise?
>
> In objectivist terms, all that matters for the sake of randomness is that
> the sequence satisfy what von Mises called "The Law of Excluded Gambling
> Systems", which is just to say (roughly) that the sequence must contain no
> predictable sub-sequences, i.e., that the result of each toss is
> independent of the others. (Subjectivists have a different way of saying
> essentially the same thing.)
>
> -gts
>
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