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SV: Capacity of the brain

Bill Todd billtodd at foo.mv.com
Mon Nov 29 15:59:28 EST 1999

Thanks for taking the time to respond.  It has been a very long time since I
studied or used any of the related physics or math, but it jogs memory
sufficiently to make it at least somewhat comprehensible.

I'm not sure that it addresses the point of defining 'random' (save possibly
implicitly) in the mathematical sense, which is what I was looking for.  I
eventually (should have much sooner) looked up an on-line dictionary
definition (posted elsewhere) that I'm comfortable with that encompasses
both the position you and others have presented and the two fumbling
attempts I made - but I remain curious as to the exact differences between
that definition and the formal one that seems to be most generally accepted

- bill

Maynard Handley <handleym at ricochet.net> wrote in message
news:handleym-2911990934450001 at handma3.apple.com...
> 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
> >of
> >> >all possible outcomes are equal?  If that is the proper definition,
> >> >'random' is a far stronger characterization than 'non-deterministic',
> >> >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.
> >> 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
> >> Gaussian.
> >
> >... which I would say makes the sum of a multi-die roll non-random,
> >non-deterministic as well.  In fact, I might go so far as to say that
> >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
> >'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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