x + y + z - t as Partial Inverse Causal Metric

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

COPYRIGHT NOTICE
x + y + z - t as Partial Inverse Causal Metric
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005

The metric:

1) dx^2 + dy^2 + dz^2 - dt^2

also known as the proper time interval squared (e.g., in reverse order
of (1), that is to say, multiplying (1) by -1), could never be
transformed to x + y + z - t, right? Well, maybe the word
"transformed" should be replaced by "represented by" in a generalized
sense, but in fact there's a way to do it.

I've proven in earlier postings to sci.stat.math and geometry.research
and elsewhere that:

2) 1 + y - x

is a partial one-sided inverse to Euclidean (and Euclidean-like)
distance of type:

3) sqrt(x^2 + y^2)

where for convenience I've left the second point (besides (x, y)) as
(0, 0). However, readers may recognize the expression above (2) as
P(A-->B) or P' (A-->B) = 1 + P(B) - P(A) with y = P(B), x = P(A).

Let's consider the expression:

4) P(A-->B) = r P(A)

Is there anything curious about this, with r a positive real number?
Well, those who know conditional probability can look at the analog
there with P(A) replaced by P(B)

5) P(B|A) = rP(B)

With r = 1, (5) is equivalent to A and B being statistically
independent. A simply has "no effect" on B in (5). Arguably, if A and
B were very dependent, then A would "come through" instead of B on the
right side of (5) to yield:

6) P(B|A)= rP(A)

with r = 1 presumably. The Probable Influence (PI) analog of this
argument is (4) above with r to be solved for.

Here's a theorem regarding (4) which readers can try to prove as
homework.

Theorem. The following holds:

7) [P'A-->B) + P'C-->B) + P'D-->B) - P'T-->B)]/20 = P(A) + P(C) + P(D)
- P(T)

holds if P(A), P(C), P(D), P(T) for sets A, B, C, D are each between
..01 and .05 and P(A) > = P(B), P(C) > = P(B), P(D) > = P(B), P(T) > =
P(B), and P(T) > = P(A), P(T) > = P(C), P(T) > = P(D). (I've typed
P'A-->B) for P'(A-->B) because my keyboard is having backward erasing
problems.)

Hint: Use the same r in (4) for P'(A-->B), P'(C-->B), P'(D-->B), and
P'(T-->B), and see what conditions are required for the remaining
inequalities to hold. Notice that P(A) < .05 is a basic definition of
a Rare Event A, although one could also choose P(A) < .01 for example.

Osher Doctorow

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