Re: P(A-->B-->C) + P(A-->C-->B) = P(A-->BC) + P{(A U B U C) --> BC}
From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 12/09/04
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Date: Thu, 9 Dec 2004 21:21:42 +0000 (UTC)
On 9 Dec 04 14:34:34 -0500 (EST), Osher Doctorow wrote:
>P(A->B->C) + P(A->C->B) = P(A->BC) + P{(A U B U C)-> BC}
I remind readers that P(A-->B), also written P(A->B), is the closest
thing that we have to "probable causation of B by A", and contrasts
with correlation. Why is P(A->B) related more to probable causation
than to correlation? The expression (A-->B) is defined as (AB')'
which is "the complement of 'the intersection of A and the complement
of B'," which is the set/event theory analog of the logical condition-
al a->b or a-->b for propositions a, b, where the latter expression
a->b is the formal version of "a implies b" (the latter is written
with a double arrow from a to b when emphasis is on the truth values
and is sometimes distinguished as "implication" ("the truth of a
implies the truth of b") as distinct from the logical conditional
where truth is not explicitly mentioned). Just as there is a deeper
relationship than "correlation" between a and b in a->b (which is
technically ~(a ^ ~b), that is to say "it is not the case that a
holds and b doesn't hold"), so there is a deeper relationship between
A, B in A->B = (AB')' = A' U B than the mere occurrence of A, B
together as a "correlation" type event or phenomenon or thing or
process.
What if we were asked to spell out the relationship between -> in
a->b and -> in A->B other than the "formal" one of following iso-
morphic or even identical laws other than the letter used and the
replacement of ~ by ' and a^b by AB (or A with an upside down U
followed by B)? The idea in a->b is that if a occurs (proposition
a), then proposition b follows. There is no reason not to
conclude the "same thing" with sets/events A, B: if A occurs, then
B follows or occurs. Whether we call this a "well-motivated
conjecture" or simply "the same idea", I don't think that anybody
other than one who has an anti-logical axe to grind will disagree.
In A->B->C compared to A->C->B, we are comparing two sequences of
causation or influence, from A to B to C and from A to C to B, and
when we attach a probability to each, we have a deeper probabilistic
expression than a mere correlation of A, B, and C or a mere sequence
of A, B, C, etc. In A->B->C, we have A influencing B and B influ-
encing C since it is defined as (A->B)(B->C) where the adjacency
of parentheses means intersection ("and"). Ordinarily we would
expect some time, even "infinitesimal", to elapse from A to B to
C so that B is not before A in time but rather A is before B and
so on, but technically they could all occur simultaneously since
a non-spurious correlation is a causation. If they do occur
simultaneously, however, the expression A->B->C selects the influe-
ence or causation of A on B and B on C rather than A on C and C
on B (the latter occurs in A->C->B). This is not terribly different
from the idea in path analysis, except that path analysis uses
regression which replaces P(A->B) by P(B/A) = P(AB) divided by P(A)
if P(A) is not 0 and there is no definition of (B/A) without the
probability, unlike (A->B) having meaning without P(A->B). Also,
P(B/A) is the probability of B with A fixed or given, which means
with A constant, unlike P(A->B) which does not require holding
anything constant as opposed to varying.
It can be concluded that P(A->B->C) type expressions are deeper than
expressions and operations in path analysis.
Osher Doctorow
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