From Osher Doctorow
COPYRIGHT NOTICE
Acceleration of Universe As Mixed Acceleration of Probable Correlation
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
Probable Correlation, not to be confused with other types of
"correlation", comes from Probable Influence (PI) Theory and is given
by:
1) P(X<-->Y)(x,y) = F(x,y) + R(x,y)
where F(x,y) is the joint cumulative distribution function (cdf) of
continuous random variables X and Y, and R(x,y) is the joint
(bivariate) reliability of X and Y. Reliability is a major
probability-statistics topic in engineering especially. Like Probable
Influence P(X-->Y)(x,y) or P(A-->B), Probable Correlation is derived
from first principles, in fact from the definition of P(A-->B) and set
and ordinarily probability-statistics and would have been discovered
long before my wife Marleen and I discovered it in 1980 had
mathematical probability-statistics researchers been as interested in
self-examination and foundations of their field and individual
applications as in governmental-industrial applications.
To see where we are heading, readers should know that the mixed partial
derivative Dxy(F(x,y) = f(x,y) where f(x,y) is the joint probability
density function (pdf) of X and y and Dxy is the partial derivative
with respect to y of the partial derivative with respect to x. In
symbols:
2) Dxy(F(x,y)) = f(x,y)
for X, Y continuous random variables. Therefore from (1) and (2) we
get:
3) Dxy[P(X-->Y)](x,y) = f(x,y) + Dxy[R(x,y)]
Dxy is not acceleration but its "cousin", sort of a "mixed
acceleration" obtained by first taking one variable, say x, and
differentiating P(X-->Y) with respect to it, and then taking the second
variable y and differentiating the previous result with respect to it
(the order of differentiation is interchangeable for continuous random
variables).
Because f(x,y) is a rather well known quantity, while Dxx[F(x,y)] is
rather poorly understood compared to it, it is arguable that equation
(3) is the PI analog of acceleration of the universe, especially if one
of the random variables such as X is a function of time T, and for
simplicity we'll look at this case first with X = T and x = t is a
value of X (respectively t).
So what does (3) tell us in terms of clues to the acceleration of the
universe? Well, the biggest difference between (1) and (3) involves
the difference between F(x,y) and f(x,y). While F(x,y) is monotone
increasing in x with y fixed and in y with x fixed, f(x,y) isn't
limited in this way but typically goes toward 0 as x --> infinity and
as x --> -infinity, and the whole graph is bounded in ordinate (height,
i.e., y) for known random variables X and Y which are continuous. So
whereas F(x,y) is asymmetric and rises to the right from 0 to 1 under
the above conditions, f(x,y) is often but not always symmetric and
falls to right and to left of its usually single population mean.
Dxy(F(x,y)) would correspond to an accelerating universe arguably if
f(x,y) changed to a "higher" pdf on the particular interval of time,
say [to, t1]. Let's call the "higher" pdf on that interval g(x,y).
Roughly but not precisely (although the rough picture can be used
provided that you follow my qualifications), X and Y are becoming "more
probable" in the interval of time considered. For example, if X is
time as a random variable (or a function of time) and Y is knowledge or
semantic information, then knowledge and time are becoming jointly
"more probable" in the interval under consideration. In semantic
information terminology, semantic information and time are becoming
more probable in the interval under consideration.
If this seems as though I am saying that the universe is "learning"
more, then provided that we don't adopt an anthropomorphic view of
"learning", this example isn't very far from what I'm saying, except
that time or some increasing function of time regarded as a random
variable is also becoming more probable.
Roughly speaking, in this interpretation of PI, the physical universe
accelerates when it probabilistically "learns more" and time gets "more
probable".
The quantity Dxy[R(x,y)] is less well known than f(x,y), and it should
be interesting to see what has been and can be learned about it in
coming years and decades.
Osher Doctorow
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