From Osher Doctorow
COPYRIGHT NOTICE
Multiplication As Summation Versus As Statistical Independence
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
The rectangular hyperbola:
1) xy = k (k constant)
is found in the Weak Heisenberg Uncertainty Principle (WHUP), in
superstring/M theory duality, and other contexts (see my recent
threads).
On the other hand, statistical independence can be shown to be
equivalent to:
2) z = xy
where z, x, y are:
3) z = P(AB), x = P(A), y = P(B)
and P(AB) is the probability of events A and B, P(A) is the probability
of event A, etc.
Equation (1) arguably represents "the whole (k) is the sum of its
parts" when k is positive. Although this seems to make sense for
positive integer k, it also makes sense for fractional and rational and
irrational k because of what is now known about fractals. It is true
that x, y, and k have to be real-valued for this to make sense. If
they are complex-valued, multiplication is no longer geometrically
increase in length or area but rotation as well, and for other fields
the interpretations can be even more compicated.
Equation (2) subject to (3) says something very, very different. It
says that the "whole" P(AB) splits into parts in a certain way across
different values or sizes of the whole! This is because, among other
things, z = P(AB) isn't constant but can change. Before we rush to
label it an invariance principle, however, notice that that
"statistically independent" things are among the most "dull" things in
the universe since they either don't influence each other or influence
each other in a rather "minimal" way from an intuitive viewpoint. Two
coins or two dice that are not loaded and which are tossed in the air
without looking or without trying to modify the outcome mechanically or
physically are considered to be statistically independent.
Yet (1) is the equation of a multiplicative inverse y = k/x for x not
0, and so y and k behave oppositely in an intuitive sense - but I just
pointed out that xy = k represents "the whole is the sum of its parts"!
What's going on?
In the equation xy = k, the whole k is a fixed quantity somewhat like
energy that is conserved, and we can regard the equation as saying that
a multiple x of y yields the whole k but also that a multiple y of x
yields the whole k. It is this two-sided decomposition of k and the
fact that k is a fixed or conserved quantity which results in opposite
behavior of x and y in the form y = k/x (or x = k/y).
In the equation P(AB) = P(A)P(B), which for A = {w: X(w) < = x} and B =
{w: Y(w) < = y} for continuous random variables X, Y becomes F(x, y) =
FX(x)FY(y) where F(x, y) = P(X < = x, Y < = y) and FX(x) = P(X < = x)
(these F's are called cumulative distribution functions (cdfs)), we
really have a "hidden" functional relationship F(x, y) = FX(x)FY(y) or
for probability density functions f(x,y), fX(x), fY(y) we get f(x, y) =
fX(x)fY(y). It is one of the simplest and least "causal" relationships
between random variables X and Y.
On the contrary, the equation xy = k is remarkably causal! We can
regard an increase in x as causing a decrease in y (for k positive) or
vice versa!
So the invariance (k) of a product (xy) can change almost non-causation
to causation! Even if both scenarios involve multiplication in similar
ways except for invariance!
But doesn't this make P(B|A) = P(AB)/P(A) more plausible causally than
P(A-->B) = 1 + P(AB) - P(A)? No. The first equation isn't a
"conserved" or constant form even though A itself is interpreted as
"fixed" or "given". It resembles more z = xy than k = xy, for z
function of x and/or y. The second equation involves subtraction, and
readers can have a homework assignment to show that subtraction is an
"encounter of a third kind" which is neither "whole is the sum of its
parts" nor "A is statistically independent of B" but rather enters into
"A exerts probable causation/influence on B". This is despite the fact
that addition and subtraction are (additive) inverses!
Osher Doctorow
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