From Osher Doctorow
The Theorem in 191.0 is proven as follows, with the slight
modifications of the Theorem indicated below in the case of (1) of
191.0.
(1). If B is a subset of A with probability 1 (w.p.1), then AB = B
(w.p.1) so P(AB) = P(B) so P(A-->B) = P(A). Therefore P(A-->B) = P
' (A-->B). But P ' (A-->B) = 1 + P(B) - P(A) with P(B) < = P(A) = 1
iff P(B) = P(A), and since B is a subset of A (w.p.1), since means
that B is a maximal such subset (w.p.1).
If A is a subset of B (w.p.1). then P(AB) = P(A) so P(A-->B) = 1. P
' (A-->B) = 1 + P(B) - P(A) with P(B) < = P(A), but since A is a
subset of B (w.p.1), we must have P(A) < = P(B) by monotonicity of
probability so P(A) = P(B), so P ' (A-->B) = 1.
(2). If A is not a subset of B (w.p.1) and B is not a subset of A
(w.p.1) but they are disjoint, then P(A-->B) = 1 + P(AB) - P(A) = 1 -
P(A) = 1 iff P(A) = 0. But P ' (A-->B) = 1 + P(B) - P(A) = 1 iff P(A)
= P(B), and the latter may or may not hold for disjoint sets depending
on the particular sets. So P(A-->B) is optimal (1) when P(A) = 0,
but there is no general optimization of P ' (A-->B).
(3) In the case of (2) above except that A and B are not disjoint (w.p.
1), this means that P(AB) > 0. But P(A-->B) = 1 + P(AB) - P(A) = 1
iff P(AB) = P(A) iff A is a subset of B (w.p.1) which is false, so
P(A-->B) cannot be optimized (at 1). P ' (A-->B) is as in (2) above.
Osher Doctorow
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