Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

The intuition behind these and other various probabilistic inequalities
is extremely important both within mathematical probability-statistics
and for applications, and it is not surprising that it escaped the
quantum theorists of the 1920s and 1930s and even up to the 1960s since
mathematicians didn't discover either many of the inequalities or the
intuition so to speak before 1960.

I have referred repeatedly to Lehmann (1967), whose reference is in my
earlier posts of this or recent threads, one of the pioneers of this
field which in the Mainstream goes by the name of statistical
dependence, statistical or probabilistic orderings, majorization, etc.


One of the best intuitive and technical presentations is by Professor
Harry Joe, Professor of Statistics, U. British Columbia, in
Multivariate Models and Dependence Concepts, Chapman and Hall: London
1997. His Chapter 2, Basic Concepts of Dependence, should be required
reading for all quantitative scientists in my opinion. Professor
Thomas S. Ferguson of UCLA told me about Harry Joe's book, and I almost
immediately ran out and bought it.

Let's look at positive quadrant dependence, which is defined by two
equivalent formulations:

1) P(X > x, Y > y) > = P(X > x)P(Y > y) for all real x, y
2) F(x, y) > = FX(x)FY(y) for all real x, y

Harry Joe points out the intuition (the discoverer was Lehmann (1967):
X and Y are more likely to be large or small together compared to
statistically independent but otherwise similar distributions when (1)
or (2) holds. Negative quadrant dependence has reversed directions of
the above inequalities.

How could anybody study statistical inequalities, such as the
Heisenberg Uncertainty Principle (HUP), without having the faintest
idea about such intuition? In my humble opinion, they can't, which is
part of the explanation of why mathematical statistics-probability
people have been so long in criticizing the HUP, although Max Jammer of
CUNY and Bar-Ilan U. was ahead of us in this respect in at least
logically and empirically and philosophically criticizing it in his
Philosophy of Quantum Mechanics, Wiley: N.Y. 1974.

With regard to the HUP, physicists can no longer claim that the HUP is
"uncriticizable" since there is now a sizeable literature on
mathematical probability-statistics inequalities, and it is growing
fast.

But even faster than the Mainstream growth is the growth of Knowledge
about PI Probable Correlation, and I want to end this post with some
comments on it.

After Joe introduces dependence, he introduces various stochastic type
orderings, but never quite captures the Rare Event essence of PI and in
particular Probable Correlation P(X<-->Y) which is still so
Non-Mainstream and little known (one has to look for it very hard) in
the literature. Probable Correlation makes stochastic type orderings
much more important than before because it gives a handle on P(X<-->Y)
= F(x, y) + P(X > x, Y > y) as a comparative tool to compare different
random variables in pairs (X, Y), (Z, W), (X, Z), (X, W), (Y, Z), (Y,
W) and in turn to compare known probability-statistics distributions to
each other with regard to Probable Correlation. As I indicated before,
comparing F(x, y) and G(x, z) or F(x, y) and F1(x, y) for different
joint cdfs F, G, F1, by means of inequalities is the key thing in
P(X<-->Y).

Osher Doctorow

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