Re: Does a "pure" real valued probability function mak

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 12/03/04 

Date: Fri, 03 Dec 2004 03:07:15 -0800

Hi,

About the notion of infinitesimal probability I'm glad that you consider
it formally acceptible.

http://groups.google.com/groups?q=infinitesimal+probability

Dave Seaman writes in the first result of that query of a/the usenet
database, comprised of the works of those who wrote in to usenet and did
not have the no-archive flag set in their post headers, about
infinitesimal probability vis-a-vis probability zero. He claims that a
random selection from the unit interval of reals has probability zero,
instead of a postive, infinitesimal probability. If x is a real on the
unit interval, then P(x), the probability of selecting x, from a uniform
random distribution, uniform in that each other real number has the same
probability of selection, Dave, representing that school of thought, has
P(x) being zero. Me, I say its probability is more along the lines of
1/2^n, where n is the unit scalar infinity.

Then, I also have the number of reals in the unit interval being 2^n, and
then, the sum of the probabilities over the entire interval is one. This
is a good feature in a probability distribution, for the sum of all
possible results is equal to unity.

If P(x) is zero, and not an infinitesimal, then the corresponding total
sum would also be zero, and then selecting a random real from the unit
interval is also zero. Pick a number between zero and one. The problem
with selecting a number is that it's not random by any means I can
consider. That's not necessarily so.

It's simple to begin to sample a real number form the unit interval: flip
a fair coin. While not necessarily "random", there are so many events and
continua involved in the physical process that it is not necessary to be
concerned of external factors that affect the coin flip. My way is to
count the number of times the coin flips by the frequency of the sound it
makes, whether it be odd or even. Anyways, flip a fair coin, you have
started sampling a real number. Assign heads and tails to 1 and 0, if the
coin flip is heads then the first (binary) digit is 1, else 0.

.0

Flip again. Now, you are sampling two numbers, the one you started
sampling with the first coin toss, which has two random digits, and the
second, with one random digit.

.00
.0

Repeat.

.100
.10
.1

Etcetera

.0100
.010
.01

That's an iterative process and unfortunately never for any finite amount
of coin flips generates the exact specification of a real number between
zero and one.

What it does show is that for any finite interval bounded by rational
numbers the denominator of which is even, that there is a specific
combinatorically determined probability that the eventually sampled number
would be in that range within the unit interval of real numbers. As well,
as each disjoint interval of the same length has the same exclusive
probability of containing a sampled value, it would appear that over all
of the unit interval that each point was as likely as each other to be
selected.

So, for any process that could select infinitely many random bits, a
question is if it generates a random real, and how many of them. From the
semi-infinite bit sequence, where semi-infinite means it has a beginning
and then trails off infinitely, that can be used as the binary
representation of a real number, and then toss the first digit of that and
then there is a completely different real number, and so on and so forth.
In that sense if the infinite sequence ended with an infinitely repeating
sequence, which seems remarkably unlikely, then the semi-infinite bit
sequence has sampled less real numbers. That is because as the sample is
retrograded or what-have-you, with the first bit removed and the result
string considered a new sample, if it represents a rational then at some
point the terminator repeats, at which point after discarding each bit of
one iteration of the repeating terminator, then no new samples are formed.

That might have to do with selecting an natural integer m at random, yet
another wild improbability, yet for some preferable as it would be of
probability 1/oo instead of 1 / 2^oo, and then an integer n less than or
equal to that. Flip the coin m times, and say the n trailing digits
repeat. That generates a variety of rationals, depending on the
distribution over the integers.

Consider if the goal was to get a distribution over the reals with some
kinds of numbers more likely to be represented in the population than
others, statistically.

If a sequence is semi-infinite, where its probability of being a rational
sequence is about as small as its probability of being an infinite
sequence, then if it doesn't represent a rational sequence then it
represents probably an aboslutely normal sequence as does each subsequence
derived by discarding finitely many bits from the beginning.

One of the problems in considering the real numbers on the unit interval
is that the regular form of expression of the number is the expansion, or
reduced fraction of integers, eg 1/2. That's a problem in the presence
of infinitesimals because there is no expansion to represent an
infintesimal or infinitesimal difference from the real number, and in many
"nonstandard" perceptions or models of the real numbers, there are
infinitesimals among the real numbers. Briefly, I describe the
infinitesimal differences form definite real numbers represented by a
convergent expression as representing definite real numbers, and the
infinitesimal differences from them indefinite real numbers. The reals
are gapless and complete and there are only and everywhere reals on the
unit interval.

There is considerable argument about the acceptance of infinitesimal
probabilities. Then again, there always has been, in some form, since the
ancients and the debate about potential and completed infinities. In some
sense that implies both. It's easy to find seeming contradictions about
the infinite, even among mathematical sophisticates, the true goal is to
extirpate them all.

Well, I guess I'll go read some more of what I and others wrote before
about infinitesimal probabilities. If I think of something good, I'll
catalog it into my mathematical ontology.

Ross