Natural Science Forum / Physics / General Physics / July 2008

From Osher Doctorow

Notice from the previous Subsection that the Gaussian/normal bivariate
probability density function is a severe "deformation" of the Star
Product because:

1) x^2 + y^2 - ksqrt(xy)

is very different from the Star Product:

2) (x^2 o y^2) = x^2 + y^2 - x^2y^2

In fact, (1) looks more like:

3) (x - y)^2 = x^2 + y^2 - 2xy ("kernel" of the Euclidean distance
function/metric)

which is in the opposite direction to Probable Causation/Influence
(PI) 1 + y - x which is in fact a one-sided partial inverse of the
Euclidean distance function.

Is this because of the quantity (x - E(X))^2 in the definition of the
univariate (marginal) Gaussian/normal distribution?   To a
considerable extent, it is because of this, which is located not in a
polynomial but in the exponent of an exponential function.

At a deeper level, (a - b)^2 for any real quantities a, b disguises
the relationship between a and b by confounding a > b and a < b
cases.   For example, (5 - 3)^2 = (3 - 5)^2 so that the case 3 < 5 and
-5 < 3 are confounded, or equivalently negative and positive values
are confounded in a distribution that is supposed to usually take on
both values.  This is "Counter-Causal."

Osher Doctorow