Correlation Probability Confidence Intervals in PI

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/11/04 

Date: Wed, 11 Aug 2004 12:02:08 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Correlation Probability Confidence Intervals in PI
Copyright by Owner Osher Doctorow Ph.D.
First Published 2004

For PI Type-C Attractors, namely A such that P(A-->B) = 1, P(A) = 0,
and P(B-->A) < 1, it is fairly easy to prove:

Theorem. a < P(A<-->B) < b iff 1 - b < P(B) < 1 - a for A a PI
type-C Attractor.

Proof. For A PI type-C Attractor, P(A<-->B) = P(A-->B) + P(B-->A)
- 1 = 1 + B(B-->A) - 1 = P(B-->A) = 1 + P(AB) - P(B) = 1 - P(B),
and therefore P(A<-->B) < b iff 1 - P(B) < b iff P(B) > 1 - b, and
similarly for the a lower bound of P(A<-->B) correspond to the P(B)
upper bound, that is to say a corresponds to 1 - a of the respective
quantities. Q.E.D.

Example. .5 < P(A<-->B) < .75 iff .25 < P(B) < .5 when A is a Type-
C PI Attractor.

Remark 1. Notice that in simple linear regression, the square of the
correlation coefficient gives the proportion of variance explained
by the independent variable, and this idea continues to the multiple
regression coefficient which uses a Pearson product moment type
correlation coefficient and so on. However, the PI Probable Correla-
tion, P(A<-->B), does not have to be squared. When converted to
random variables, P(X<-->Y) for continuous random variables is the
probability of X if and only if Y with A = {w: X(w) < = x}, B =
{w: Y(w) < = y}, and indicates the probability that X and Y "vary
together" in the inequality sense which is familiar to statisticians
in dependence theory if not elsewhere. This is surely a good reason
to refer to it as a "probable correlation," remembering that correla-
tion is co-relation. It also incorporates the same direction on
inequalities. The objection that nothing corresponds to negative
correlation is easily solved: simply calculate P(AB') or P(A'B).

Remark 2. From Remark 1 applied to the above Example, P(A<-->B)
between .5 and .75 would correspond to linear correlation or linear
regression having correlation r between .70 and .86 roughly and to
X accounting for between 50% and 75% of the variation in Y. This
occurs as B varies between .25 and .50, which is conveniently on one
side of the mean Fairly Frequent Event probability of .50 and well
within the Fairly Frequent Event range of .05 to .95. It is
interesting that the upper half of the range, .50 to .75, for P(B)
is not at all near optimality for P(A<-->B), and this upper half of
the range is closer to the Independent Probability-Statistics (IPS)
domain of .95 to 1.00.

Remark 3. Notice that P(A<-->B) varying with P(B) has a linear-
like character, even though P(A<-->B) = P(B) would be PI Indepen-
dence (although there is the better version of PI Independence in
which P(A-->B) is simply changed from 1 + P(AB) - P(A) to 1 +
P(A)P(B) - P(A)). In fact, P(A<-->B) could be said more accurately
to vary with 1 - P(B), which although linear is decreasing in P(B).
This suggests exploring P(A<-->B) = 1 - P(B). The latter does not
work, but related inequalities might well do so. The BCP version
P(B/A) = P(A) (and in fact also P(B/A) = P(B)) always have a "linear-
like" character. Does this hint that linearity may be an "illusion"
(or a way of looking at things) of Fairly Frequent Events but only
toward the Rare Event boundary? And that Rare and Very Frequent
Events are in some sense nonlinear? (By the way, P(A<-->B) seems
to cross linearity and nonlinearity, unlike the Pearson Product-
Moment and related coefficients.)

Osher Doctorow

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