Re: Statistical Dependence in PI vs Conditional Probability/Bayesian

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

The "Non-Dependence" of X and Y in PI is, from the second posting back:

1) P(X-->Y)(x, y)_IND = 1 + FX(x)FY(y) - FX(x)

This is not the "independence" or "statistical independence" of X and Y
in PI or in anywhere else. "Independence" in probability-statistics is
not measured on a continuous scale but on an "all or nothing" scale, as
some 0 level of dependence (DEP) measured in some continuous way or
simply as the applicability of the equation F(x, y) = FX(x)FY(y), which
applicability is likewise not continuous but "all or none". We have
seen that dependence (DEP) is differently measured in PI and in
conditional/Bayesian probability-statistics.

To recapitulate the situation for F(Y|X=x)_IND, it is:

2) F(Y|X=x)_IND = FY(y)

Whatever it is that F(X-->Y)(x,y)_IND measures, it is at least as large
in magnitude as F(Y|X=x)_IND, because:

3) 1 + FX(x)FY(y) - FX(x) > = FY

since (3) is equivalent to:

4) 1 - FX(x) > = FY(y)(1 - FX(x))

Inequality (4) is true because if FX(x) is not 1, then it just says 1 >
= FY(y) = P(Y < = y) which is true since all probabilities are between
0 and 1. If FX(x) = 1, then (4) just says 0 > = 0.

Intuitively speaking, objects or expressions or processes take on an
_IND subscript when they "split" or "separate as variables"
multiplicatively according to F(x, y) = FX(x)FY(y) or an analogous
equation. This is a dangerous warning concerning solving partial
differential equations (PDEs) by separation of variables, but that
isn't the main intent of this thread or this part of the thread anyway.
The most important point is that "independence" as obeying F(x, y) =
FX(x)FY(y) is one type of all-or-none "thing", while the effect of that
type of thing on continuous measures such as P(X-->Y)(x,y)_IND or
F(Y|X=x)_IND is in fact no longer an all-or-none "thing" but the
particular continuous measure itself.

Osher Doctorow

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