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SV: Capacity of the brain

Maynard Handley handleym at ricochet.net
Mon Nov 29 12:34:45 EST 1999


In article <81pcoa$gub$1 at pyrite.mv.net>, "Bill Todd" <billtodd at foo.mv.com>
wrote:

>Matt Jones <jonesmat at ohsu.edu> wrote in message
>news:81mnkr$j06$1 at fremont.ohsu.edu...
>> In article <81jmfs$fv2$1 at pyrite.mv.net> Bill Todd, billtodd at foo.mv.com
>> writes:
>> >Am I the only one who believes that 'random' means that the probabilities
>of
>> >all possible outcomes are equal?  If that is the proper definition, then
>> >'random' is a far stronger characterization than 'non-deterministic', and
>> >some of the foregoing discussion points may have been flying past each
>other
>> >without contact.
>> >
>>
>> Think of the classical example of something "random": the Gaussian
>> distribution. This is definitely NOT a case of all probabilities being
>> equal. There is a peak in the distribution at which the probability is
>> higher than at other values. What you are thinking of is a "uniform"
>> distribution, such as when you flip a fair coin or roll a fair die. Note
>> that if you roll two fair die, and calculate the probabilities of their
>> SUM, you no longer have a uniform distribution, but something closer to a
>> Gaussian.
>
>... which I would say makes the sum of a multi-die roll non-random, though
>non-deterministic as well.  In fact, I might go so far as to say that even
>the result of a one-die roll is not random, and amend my (unfortunately
>mis-)stated understanding of 'random' to mean 'unpredictable in any way,
>including probabilistically':  otherwise, one could state that virtually
>every physical phenomenon exhibits random behavior, which is not
>particularly useful.
>
>So what I'm asking is whether there is indeed a well-accepted definition of
>'random' in this context, and if so what it is.

Any mathematical discussion of probability makes it quite clear that their
starting point is a measure defined on some space, and the measure is
something "god-given" and outside the realm of mathematics. Once you have
a measure, you can modify it in various ways to generate other measures
corresponding to the behavior of functions of random variables, but the
starting point is outside the theory. 

Depending on how you look at it, you could argue that this is in fact seen
in practice. As far as statistical mechanics is concerned, in a system at
thermal equilibrium energy is nominally spread across all the distinct
states of the system weighted by Exp[-kT]. However how does one count
states of a system as being distinct? At a more abstract (and
mathematically elegant) fashion, one describes the system in a new way
that bunches apparently distinct boson states together as all being the
same state (Fock space and all that). In a more elementary scheme, one
considers the apparently distinct boson states as actually distinct and
thus munges the measure so that each state gets a smaller weighting.

Finally Gaussians are an interesting case in that one can prove (central
limit theorem) that subject to certain not-very-onerous conditions on the
underlying measure, a random variable that consists of the many many sums
of other random variables will be distributed according to a Gaussian.
This is why, for example, errors in measurement are assumed distributed
that way---we assume the actual error is a sum of many different
uncorrelated influences. For the same reason, financial instruments are
assumed to vary according to a Gaussian, however in that case it's not
clear that this is valid---the instruments vary most of the time according
to the many sums of small uncorrelated effects, but every so often
something big comes along that swamps all that. 

The point is that, you can go a long way once you accept the mathematical
framework. When it comes to generating input for the mathematical
framework you leave mathematics and the discussions become a lot more
vague.

Maynard




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