From Osher Doctorow
For example, let's look at:
1) P(A-->B) = 1 + P(AB) - P(A)
When A and B are independent or statistically independent, then:
2) P(AB) = P(A)P(B)
and (1) becomes:
3) P(A-->B) = 1 + P(A)P(B) - P(A) = 1 - [P(A)(1 - P(B))]
It is useful to express the left hand side of (3) as P(A-->B)_IND
where IND stands for "independent", and we have:
4) P(A-->B)_IND = 1 - [P(A)(1 - P(B))]
But the second term on the right is DEP_Log(B, A), so:
5) P(A-->B)_IND = 1 - DEP_Log(B, A)
Since Independence is both intuitively and actually a "low Causation"
scenario, (5) agrees with the idea of the previous post, namely to use
DEP_Log rather than DEP_2 Log in such cases.
By the way, we also have:
6) DEP(A, B) = P(A-->B) - P(A-->B)_IND
which again shows the fundamental nature of P(A-->B) even in
(statistical) dependence.
In physics, in which interaction and dependence of objects is very
important usually, the use of either dependence or independence
(equations (6) or (2) as illustrated by (4) and (5) respectively)
without using or even understanding the underlying structure in terms
of P(A-->B) is arguably a mistake. Notice that (6) says that
dependence is the non-independent part of Probable Causation/
Influence, where intuitively two things are independent if neither
affects the other. This agrees with our intuition and is far more
precise than the use of "Causation" in theoretical physics as limited
to light cones (in fact, in the latter context, only "bounds on
Causation" are really being studied, and even those are arguably
modifiable).
Osher Doctorow
Osher Doctorow
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