Quantum Gravity 347.3: When Does P ' {(A ' --> B)

Subject:	Quantum Gravity 347.3: When Does P ' {(A ' --> B) \| A ' )} > P(B) ?
Date:	Sat, 5 Dec 2009 21:58:00 -0800 (PST)

>From Osher Doctorow

In mainstream mathematical probability/statistics, "Conditional
Probability" (CP for short here), of which one type is Bayesian
Probability (BP for short here) is defined by:

1) P(B|A) = (definition) P(AB)/P(A) if P(A) is not 0, where / on the
right is division of real numbers.

However, Probable Causation/Influence (PI) has shown that there is a
modification or alternate named P ' (B|A) defined by:

2) P ' (B|A) = P(B)/P(A) if P(B) < = P(A) and P(A) is not 0.

For example, if B is a subset of A with probability 1, then (1) and
(20 are equal.

I will prove here:

3) Theorem. If P(A<-->B) = 1 + DEP(A, B) and P(A) > 1/2, then (4)
holds below:

4) P ' {(A ' --> B | A ' ) } > P(B).

Remark. The significance of (4) in Conditional Probability is that it
can be shown that A and B are Statistically Independent iff P(B|A) = P
(B) when P(A) is not 0, so that P(B|A) > P(B) is positive Statistical
Dependence of A and B. (4) is part of a generalization of this to
PI.

Proof of (3). P(A<-->B) = P(AB) + P(A ' B ' ) = 1 + DEP(A, B) = 1 + P
(AB) - P(A)P(B) implies that P(A ' B ' ) = 1 - P(A)P(B), but P(A ' B
' ) = 1 - P(A U B), so P(A U B) = P(A)P(B). But we know that P(A U B)
= P(A ' --> B), so P(A ' --> B) = P(A)P(B). Dividing both sides of
the last equation by P(A ' ) yields:

5) P(A ' --> B)/P(A ' ) = [P(A)/P(A ' )]P(B)

The left-hand side of (5) is P ' {(A ' --> B) | A ' ) }, and if P(A) >
1/2 then P(A)/P(A ' ) = P(A)/(1 - P(A)) > 1 (because P(A)/(1 - P(A))
> 1 is equivalent to 2P(A) > 1 which is equivalent to P(A) > 1/2).
Therefore, (4) follows from the assumptions. Q.E.D.

Thus, a generalized notion of Statistical Dependence in the mainstream
Probability/Statistics sense holds if P(A) > 1/2 and if P(A <-->B) = 1
+ DEP(A, B) where DEP(A, B) is the non-generalized mainstream
Statistical Dependence, where P(A<-->B) is the Probable Correlation in
PI which plays in PI the role of "Dependence" in a simultaneous sense.

Osher Doctorow

Quantum Gravity 347.3: When Does P ' {(A ' --> B) | A ' )} > P(B) ?