Independence Further Examined in PI

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 01/13/05 

Date: Thu, 13 Jan 2005 13:21:07 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Independence Further Examined in PI
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005.

Definition 1. Let P(A-->B)_I be PI when A, B are statistically
independent, that is to say when P(AB) = P(A)P(B). Then we
have:

1) P(A-->B)_I = P(AB) - P(A) + 1 = P(A)P(B) - P(A) + 1

Here's another interesting definition (notice that mathematical
statisticians have mostly used "dependence" in orderings, that is
to say inequalities, rather than defining it for use in equations)
that is new.

2) DEP(A,B) or DEPENDENCE(A, B) = P(AB) - P(A)P(B)

When DEP(A,B) > 0, it corresponds to positive quadrant dependence
in the terminology of Lehmann in the 1960s when A = {w: X(w) < =
x}, etc.

The biggest surprise is:

Theorem. P(A-->B) = P(A-->B)_I + DEP(A,B)

Proof. P(A-->B)_I = P(A)P(B) - P(A) + 1, and DEP(A,B) = P(AB) -
P(A)P(B), so their sum is P(AB) - P(A) + 1 = P(A-->B). Q.E.D.

Remark. The Theorem shows that PI (Probable Influence) decomposes
into a dependent part DEP(A,B) which is plausibly "(positive)
statistical dependence") and a statistically independent version
of itself. The entire field of stochastic and statistical inequal-
ities (which has an enormous literature including that of Olkin,
Marshall, etc.) thus really belongs in or close to PI (the types
involving expectations are further removed from PI).

Osher Doctorow

Relevant Pages