From Osher Doctorow
Statistical Dependence comes in various types, but the main definition
is based on E. Lehmann's "positive quadrant statistical dependence"
from the 1960s, which when formulated in terms of probabilities of
sets/events says:
1) P(AB) > P(A)P(B) (positive statistical dependence)
From this, we can see that statistical dependence as a quantitative
variable can be defined by:
2) DEP(A, B) = P(AB) - P(A)P(B)
with statistical independence occurring when DEP(A,B) = 0 or
equivalently P(AB) = P(A)P(B).
Readers can try finding research papers on statistical dependence on
the internet, but it will probably be faster to read Annals of
Statistics and similar theoretical journals in large university
research libraries (especially mathematics-engineering research
libraries). You can also enter keywords "dependence", "statistical
dependence", and so on in sci.physics in the search box or enter my
name there and read my previous threads on the topic (and some of my
threads in sci.stat.math from past years).
We can prove the following Theorem:
Theorem. We have:
3) P' (A-->B) = P(A-->B) + DEP_Non-logistic(A, B)
4) DEP(A, B) = DEP_Logistic (A, B) - DEP_Non-logistic (A, B)
Here DEP_Logistic(A, B) is defined as:
5) DEP_Logistic(A, B) = P(B)(1 - P(A))
Notice that (5) makes sense because the Logistic differential equation
has the form:
6) dy/dt = ky(1 - y)
and the generalization of the right hand side to two variables x, y
is:
7) dy/dt = ky(1 - x)
which is the type of thing that occurs in the Lotka-Volterra
equations.
I leave the proof of the Theorem as homework for readers, just noting
that DEP_Non-logistic (A, B) turns out to be P(A' B).
Statistical dependence has gone in some rather curious directions in
the mainstream literature, including especially inequalities and
complicated mixtures rather than the simpler fundamental type of
equations here, especially since the Mainstream has no concept of P(A--
B) and P' (A-->B).
Osher Doctorow
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