From: Osher Doctorow [log in to unmask], Mon. June 25, 2001 4:37PM

I have just proven on [log in to unmask] a double probabilistic
inequality which says essentially the following.

1. Use Bayesian methods (Bayesian conditional probability or BCP) for fairly
common/fairly frequent events.
2. Use Logic-Based Probability/Statistics (LBP) for rare events.
3. Use Independent Probability/Statistics (IPS) for very common/very
frequent events provided that the events are described by cumulative
distribution functions of positively dependent random variables.

Positively dependent random variables are ones for which P(X < = x, Y < = y)
> = FX(x)FY(y) where FX(x) = P(X < = x), FY(y) = P(X < = y) for all values
x, y.  This intuitively means that X and Y (randomly) increase together or
in the same direction.

The proof can be found on anzap-l, and is part of Memory (M) Theory, which
is a generalization of my paper in B.N. Kursunuglu (Ph.D. Cambridge
University under P. Dirac), S. L. Mintz, and A. Perlmutter, Quantum Gravity,
Generalized Theory of Gravitation, and Superstring Theory-Based Unification,
Kluwer Academic/Plenum: N. Y. 2000, 89-97.  It is probabilistically fairly
simple and short, although I could not have derived it without an
understanding of fuzzy multivalued logics including Lukaciewicz (L),
Product/Goguen (P), and Godel (G).  See P. Hajek (Czech Republic).,
Metamathematics of Fuzzy Logic, Kluwer: Dordrecht 1998 for exceptionally
clear and concise comparison of L, P, G.  Any two of L, P, G together
constitute Generalized Boolean Logic.  L may be more familiar to some
mathematical logicians via its extension Rational Pavelka Logic.

Some members of ornet may be interested in what Memory (M) says in brief.
It defines Memory (M) events as those influenced by or depending on two or
more previous times, Semi-Memory (S) events as those influenced by one
previous event, and Non-Memory (N) events as those influenced by no previous
events.  M events are exemplified by viscoelastic materials (polymers,
plastics, colloidal suspensions, etc.), human memory, quantum entanglement,
many economic and biological processes, radiation, expansion/contraction of
the universe, etc..  S events are exemplified by Markov chains and Markov
processes and differential equations depending on only one previous time
(the most common type taught, with just derivatives or partial derivatives
and time dependent variables with one time t indicated - the derivative is
theoretically supposed to be at an infinitesmally tiny future time since it
involves t + h in the limit).   N events are events that are usually
referred to as independent, as in bombardment of dust particles by molecules
from random independent directions.  If two events are unrelated, they are
often considered to be independent.

Because of the above paragraph, we can actually decide on the basis of how
many previous points are involved in time influence whether to use the LBP,
BCP, or IPS models.  It should be noted that M events, which are influenced
on two or more previous times, are quite often global.  This is partly
related to the tensor nature of events which depend on more than one (and
typically 4 or more) time or spatial point.  Dr. G. Gamow's *One, Two,
Three, Infinity* begins to look more plausible from this viewpoint.  S
events are local.  Markov chains may look as though they go back more than
one time point, but in fact they are just built up from one-step-at-a-time
chains, and each link in the chain is not even a continuous interval but a
jump from one fixed point to the next fixed point in the chain.  The same
idea holds for Markov processes even though they look outwardly connected.
Independent processes cross over both global and local events or can stay in
either type.

Osher Doctorow Ph.D.
Doctorow Consultants
Formerly (and still intermittently in part) California State Universities
and Community Colleges, U.S. Defense Department, etc.