From Osher Doctorow
The remaining question is why 1 minus an expression corresponds to a
positive expression between E(Y-->X) and P' (A-->B), as for example
E(X|Y=y) of the former corresponds to 1 - P(A|B) of the latter.
Again, the simplest explanation is from the logistic differential
equation:
1) dy/dt = ky(1 - y)
where dy/dt, the "causal" variable from the viewpoint of both Birkhoff
Causation and Probable Influence/Causation (PI), depends on both y and
1 - y. The y dependence represents growth or decay/contraction
without conditions, but the 1 - y dependence represents "supply-
limited" growth or decay, which is to say that the "upper
boundary" (maximum sustainable population in biology, for example)
exerts a downward influence while the lower boundary (0) exerts an
upwards influence in the case of growth/expansion for example.
Expressions like 1 - P(A|B) or 1 - fY(y) are the analogs of 1 - y in
(1), while expressions like E(X|Y=y) and I(x)dx are analogs of y in
(1), except that in PI and Birkhoff Causation outside a purely
differential equation context both y and x may be weighted in a more
general equation than (1), for example dy/dt = kx(1 - y), etc. This
is in fact what happens somewhat in the Lotka-Volterra equations that
are generalizations of the logistic and the Riccati Differential
equations.
Osher Doctorow
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