Invariance-Intersection Theory "Collapses" Into PI 2: hP(X-->Y)(x,y)
- From: "OsherD" <mdoctorow@xxxxxxxxxxx>
- Date: 9 Sep 2005 07:04:05 -0700
>>From Osher Doctorow mdoctorow@comcast,net
Now let's look at:
1) hP(X-->Y)(x,y) = h[1 + F(x,y) - FX(x)]
and recall that:
2) FX(x) = lim F(x,y) where limit is as y --> infinity
from undergraduate probability. However, (2) is also:
3) FX(x) = lim F(x, y + 1/h) as h --> 0+
Therefore from (1):
4) hP(X-->Y)(x,y) = h[ 1 + F(x,y) - lim F(x, y + 1/h)]
But h = 1/(1/h), so we can rewrite (4) as:
5) hP(X-->Y)(x,y) = [1 + F(x,y) - lim F(x, y + 1/h)]/(1/h)
Now let's recombine terms in (5) and use 1/(1/h) = h:
6) hP(X-->Y)(x,y) = h + [F(x,y) - lim F(x, y + 1/h)]/(1/h)
where limit is as h --> 0+.
The second term on the right hand side of (6) is:
7) [F(x, y) - lim F(x, y + 1/h)]/(1/h)
where limit is as h --> 0+ or 1/h --> infinity. Although h is going in
the opposite direction to a (partial) derivative h, it is still
meaningful to compare (7) to:
8) [F(x,y) - F(x, y + 1/h)]/(1/h)
which is the secant line slope as usual.
Let's compare this to:
9) Dy(F(x,y)) = lim [F(x, y + h) - F(x, y)]/h as h --> 0
and where Dy(F(x,y)) is a symbol for the partial derivative with
respect to y of F(x,y) at (x,y). Readers surely realize that the
symbol on the left hand side of (9) merely is "shorthand" for the right
hand side of (9) which latter is the important expression. But if we
take the limit in (9) as h --> infinity instead of h --> 0, there has
been up to now no name for that expression since there seemed to be no
application of it.
It's time to give a name to:
10) lim [F(x, y + h) - F(x, y)]/h as h --> infinity
Let's call it the "Probable Causal Derivative", symbol Dc(F(x,y)) since
the symbols Dy and Dx already are reserved respectively for the partial
derivatives with respect to y and x respectively. The symbol Dc isn't
meant to refer to c varying - c stands for "causal".
The expression (10) is the limit as h --> 0+ of expression (8), but (8)
should be compared with (7). If we put a limit as h --> 0+ on the
F(x,y) term of (7), it still remains F(x,y). So the limit comes out of
the whole expression (7):
11) lim [F(x, y) - F(x, y + 1/h)]/(1/h) as h --> 0+
This is equivalent to (10) except that I've multiplied the numerator of
(10) by -1. Rather than change the order of the terms now, I'll say
that up to -1, (11) and hence (7) is Dc of F(x, y). So it can be
approximated by:
12) [F(x, y) - F(x, y + 1/h)]/(1/h)
But (12) is the slope of the secant line "going in the opposite
direction from the tangent line".
We conclude that Probable Influence/Causation P(X-->Y)(x,y) times h is
approximately the slope of the relevant secant line "going in the
opposite direction from the tangent line".
Osher Doctorow
.
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