Quantum Gravity 272.6: A Jacobson Radical Expression That Measures Statistical Dependence Via Probable Causation/Influence (PI)

From Osher Doctorow

THEOREM. Let yox = y + x - yx and DEP(A, B) be the statistical
dependence of A and B (set/events in the probability space) defined
as:

1) DEP(A, B) = P(AB) - P(A)P(B), where y = P(AB), x = P(A), z = P(B)
(for later use)

Then we have:

2) xoz - yox - yoz = DEP(A, B) - P(AB)[1 + P(A ' ) + P(B ' )]

(which, it should be noted, is a negative-valued expression on both
sides of (2)).

PROOF. By simple algebra:

3) yox + yoz - xoz = y + x - yx + y + z - yz - [x + z - xz] = y(1 - x)
+ y(1 - z) + xz
= y(2 - x - z) + xz

which is nonnegative since 0 < = x < = 1 and 0 < = z < = 1 and x, y, z
are in [0, 1] by definition of probability. Rewriting (3) using
probability notation including (1) yields the required result, noting
that P(AB) comes in as an extra factor with coefficient 1 in (2) from
the first equation of (1).
Q.E.D.

REMARK. Notice that the expression xoz - yox - yoz of (2) has only
Causes x and z in the first term, and has one Effect (y) and one Cause
(x or z) in each of the other terms. (The reason for x and z being
Causes is that P(A-->B) = 1 + y - x for example, with y = P(AB) and x
= P(A), has Cause A represented by x = P(A), and the remaining term y
= P(AB) represents the contribution of Effect B. Similarly for P(B--
A) = 1 + y - z where y = P(AB) and z = P(B).) So since xoz, yox, and
yoz can easily be proven to be nonnegative (for example, xoz = x + z -
xz and x, z are in [0, 1] by definition (see (1)) and xz < = x because
the product of two numbers in [0, 1] is less than or equal to either
of them), the Theorem shows that we can identify the Causal term of
the expression xoz - yox - yoz as the term with the positive sign (x o
z) and the Effect or mixed terms as the terms with negative signs (y o
z, y o z). This is opposite to the situation in P(A-->B) which
reduces to 1 + P(AB) - P(A) where the negative signed term P(A) is
Cause and the positive signed P(AB) represents Effect.

DEFINITION. The Triple Star Jacobson Product will be defined here as:

4) x*y*z = (definition) x o z - y o x - y o z (also written xoz - yox
- yoz)

So the conclusion of the Theorem can be written:

5) x*y*z = DEP(A, B) - P(AB)[1 + P(A ' ) + P(B ' )]

Clearly, x*y*z increases with DEP(A, B) which is just P(AB) -
P(A)P(B), because the second term of the right hand side of (5)
doesn't affect DEP(A, B) as Readers can prove for themselves (noting
that P(A), P(B), and P(AB) are "independent variables" in the usual
calculus/analysis language with time subscripts like P(A_t)
suppressed), since for two sets/events A, B, whether they intersect or
not and the probability P(AB) of their intersection and their separate
probabilities P(A) and P(B) are not determined by any two out of the
three of these terms.

Had we chosen y o x + y o z - x o z of (4) to analyze, we would have
found that it decreases with DEP(A, B). So Statistical/Probabilistic
Dependence is properly determined by the sign of the Causal term x o z
rather than the mixed or Effect terms y o x, y o z.

Note that many Readers may be familiar with the continuous random
variable form of statistical dependence, namely f(x, y) does not equal
fX(x)fY(y) for probability density functions f(x, y), fX(x), fY(y),
which have cumulative distribution function forms F(x, y), FX(x),
FY(y) respectively. But FX(x) for example = P(X < = x) = P(A) for A
= {w in probability space such that X < = x}, etc. Lehmann in the
late 1960s, in Annals of Statistics to my recollection, defined the
main measure of Statistical Dependence used now, which he called
Positive Quadrant Statistical Dependence F(x, y) = P(AB) > =
FX(x)FY(y) = P(A)P(B), so that P(AB) - P(A)P(B) > 0. When P(AB) =
P(A)P(B), then we have Statistical Independence. The reverse sign is
"Negative Quadrant Dependence".

Osher Doctorow
.

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