Conservation of Frequent Events But Not Rare Ones

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 10/06/04 

Date: Wed, 6 Oct 2004 01:18:39 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Conservation of Frequent Events But Not Rare Ones
Copyright By Owner Osher Doctorow Ph.D.
First Published 2004.

 From my thread on invariance measures today and yesterday (the third
part will appear by the time this does, I think), Fairly Frequent
Events and Rare Events respectively follow:

1) P(B/A) = P(S(A, B)) = P(A) for A an Attractor in Frequent Events
2) P(A-->B) = P(T(A,B)) does not = P(A) for any Rare Event A

Here S(A,B) and T(A,B) are respectively defined by the transforma-
tions:

3) S(A,B) = P(AB)/P(A) if P(A) is not 0
4) T(A,B) = 1 + P(AB) - P(A)

For invariant measures u or P (P used for probabilities or condition-
al probabilities with P(A) fixed), S or T should be replaced by
their inverse functions and (3), (4) generalize the concept to
transformations on two arguments (both range and domains either set-
valued or pairs of set values). Other than these considerations or
modifications, however, some similarities should be noted.

This is especially important because of Noether's Theorem which
essentially says that invariances of physical systems with respect
ro translation, rotation, and time respectively give rise to
conservation of linear momentum, angular momentum, or energy, and
invariance with respect to general gauge transformations yields
conservation of electric charge, etc. Translation, rotation, and
time are types of transformations.

What would be an intuitive interpretation of Fairly Frequent Events
being conserved under conditional probability P(B/A), while Rare
Events are not conserved under Probable Influence P(A-->B) (except
under the Universe where P(A) is replaced by 1 for the latter!)?
The simplest interpretation is that Fairly Frequent Events B can come
under the influence of "finite" Attractors in the sense of P(A) < 1,
which is a sort of analog of "Materialism" as placing high utility
on finite appearances. Rare Events B cannot come under the influence
of such Attractors and only can be attracted by the "Universe" up to
probability 0, that is to say P(A-->B) = 1 is the only possible case
of Attractors for Rare Events. This would be a sort of analog of
"Non-Materialism" as placing high utility on infinite or indeed
universal "essences". Philosophically, one could regard this as
the Hedonist versus Platonic viewpoints respectively. But this is
only part of the story. When Fairly Frequent Events are attracted
to an Attractor, they tend to "stay still" or be conserved more or
less. However, Rare Events to tend expand or increase without bound,
on the one hand, or to decrease toward negligibility (the null
set) on the other hand. If the formulation P(A-->B) = 1 is
correct, the expansion or increase is by far the dominant mode
at least for "Non-Material" variables like Knowledge - a type of
"Non-Material" principle which balances out the law of thermo-
dynamic increase in entropy for "Material" objects.

Osher Doctorow