Re: Invariance-Intersection Theory "Collapses" Into PI

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

Actually, I think I've figured it out. Witten really does his research
at Stanford, which has cool, clean air. CalTech is a cover. Well,
maybe.

Now let's get back to work.

Take a look at:

1) dy/dx = f' ' (x) = lim [f(x + h) - f(x)]/h (lim as h --> 0)

Now take a look at:

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

According to Garrett Birkhoff of Harvard, the grand old mathematician
of Harvard who is no longer with us, equations involving things like
dy/dx or partial derivatives, called of course differential equations,
embody causation. I call that Birkhoff Causation. Most people don't
call it anything. Typically Birkhoff was thinking x = t (time), but
heck - space can substitute in a pinch.

Now set A or A(x) or A(x, y) = {w: X(w) < = x}, B or B(y) or B(x,y) =
{w: Y(w) < = y}, in which case we get:

3) P(X-->Y)(x,y) = 1 + P(X < = x, Y < = y) - P(X < = x)

= 1 + F(x, y) - FX(x)

where F(x, y) is the joint cumulative distribution function (cdf) of
random variables X and Y (taken continuous here for simplicity) and
FX(x) is the (marginal) cdf of X.

So now compare (1) and (3). Well, there's something vaguely similar
about them, except for things like h. But heck, what's a little
difference among friends?

Now compare (1) and (2) again. Remember, (1) says P(A-->B) = 1 + P(AB)
- P(A).

Let's suppose that we're teaching a class here (not far wrong, except
that this may usually be a classroom of one, although it's one of my
best classes :>)

What's right about this picture? Or more precisely, what's similar
about (1), (2), and (3)?

OK, there's a difference involved. Actually, the difference is
fundamental. In fact, if you look very closely at (1), you may be
reminded that now that q-arithmetic is with us (look under that or
similar keywords on the internet), the limit part of (1) may not be
that much more important than the difference ratio. Or in geometric
language, the secant line/segment may not be that far behind the
tangent line.

Oh, oh, a student raises his/her hand! "But there's no ratio in (2) or
(3)!" No, there isn't. Well, not exactly. Well, maybe there is.
These are probabilities, remember. And the Frequency Theorists of
probability-statistics think that everything is ratios or limits of
ratios in probability-statistics. And with our Grand Secant
perspective, we can approximate, can't we? (I wonder, could one form a
new Religion using the Grand Secant persepctive :>)

So we come to the topper of this post:

4) P(A-->B) = (approximately) 1 + N(AB)/h - N(A)/h

where 1 could be written h/h, but let's not get carried away. What's
the h? Well, I'm trying to approximate P(AB) and P(A) by relative
frequencies, so I suppose that I should write N(AB)/n and N(A)/n (or
N(A)/n2) where N(AB) is some estimate of the number of events in some
finite representation of AB and n is some total population, etc. So h
= n. But h --> 0 isn't n --> infinity. Well, nobody said it would be
easy. Anyway, don't worry about the limit since we're way ahead of
where we were a little while ago.

You can see now that the 1 in (4) doesn't change the nice fact that
Birkhoff Causation from differential equations has a remarkable
resemblance to Probable Influence/Causation (PI). Anyway, you can
write 1 as h/h.

Could you do this with conditional probability P(B|A)? No. It's
P(AB)/P(A) for P(A) not 0. It's like trying to put water into a Klein
bottle in our spacetime. If I were one of the judges in Hitchhiker's
Guide to the Galaxy, I would outlaw P(B|A) entirely after this
demonstration. Don't nobody move :>)

Osher DOctorow

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