From Osher Doctorow
We know that:
1) P' (A-->B) = P(A-->B) + P(A' B)
2) P(A<-->B) = P(AB) + PA' B' )
From the first equation, if B is a subset of A or if B = A (up to a
set of probability 0), then P(A' B) = 0, and so P' (A-->B) = P(A--
B). But if A and B just overlap partly, that is to say intersect
partly, then P' (A-->B) and P(A-->B) are not the same in magnitude.
Thus, two different scenarios (physical scenarios in practice) are
described by (1), namely scenarios of (a) partial overlap of two set/
events, and (b) one set/event completely contained in the other.
From the second equation, it may not look at first glance as though
P' (A-->B) is involved, but recall that if B is a subset of A, then AB
= B, and A' is a subset of B' so that A' B' = A', both up to (that is
to say, not including) sets of probability 0. Therefore in the same
cases as in (1), if B is a subset of A or equals A, then (2) reduces
to:
3) P(A<-->B) = P(B) + P(A' ) = P(B) + 1 - P(A) = 1 + P(B) - P(A) =
P' (A-->B)
But P(A<-->B) is not just any old probability, but rather it is
Probable Correlation (defined pointwise rather than by any average in
the random variable representation, unlike Mainstream correlation
coefficients). So the split between partly overlapping and
completely overlapping scenarios has major effects at the levels of
Probable Correlation according to (2) and (3).
We arguably have plenty of motivation to call these two scenarios "two
phases", and what is at least as interesting is that we have a very
simple and intuitive model for how the phase transition occurs.
That is as follows. Take Venn diagrams of set/events A, B, each in
general a different size, and let them first be disjoint or mutually
exclusive and then move partly toward each other until they
intersect. This partly intersecting scenario is phase 1. Then take
the smaller set or if they're both of the same size either set, and
increase the intersection by moving this smaller set into the bigger
one so that the smaller one is "completely overlapped" by the bigger
one. This is phase 2.
Osher Doctorow
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