Science > Physics > Causation/Causality, Memory, and Convolution 10: Dependence Kernel Measures
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
21 Feb 2006 01:04:50 AM |
| Object: |
Causation/Causality, Memory, and Convolution 10: Dependence Kernel Measures |
From Osher Doctorow
I pointed out earlier in this thread that mathematical
probability-statistics people missed the opportunity to measure
(statistical) dependence and satisfied themselves with just studying
order relations involving dependence, but there is actually a
super-complicated branch of mathematical probability-statistics,
"dependence kernel measures", which tries to measure (statistical)
dependence the hard way conceptually though with canned computer
programs most of them don't really have to know what they're doing to
use the programs.
In contrast to Probable Influence/Correlation (PI), which concisely and
succinctly and intuitively defines the dependence measure by:
1) DEP(A, B) = P(A-->B) - P(A-->B)_IND = P(AB) - P(A)P(B)
2) DEP(X, Y) = P(X-->Y) - P(X-->Y)_IND = F(x, y) - FX(x)FY(y)
(see the earlier postings in this thread), so that in fact DEP(X, Y)
measures precisely the Lehmann (1966-1967) positive quadrant
statistical dependence and the Lehmann negative quadrant statistical
dependence, the "dependence kernel measure" people dwell in the rather
odd world of feature extraction, (energy) landscapes, spectra and the
whole frequency-domain machinery, Hilbert-Schmidt norms, etc.
This is not to say that pattern recognition and clustering and all that
is useless, but it reminds me of algebra as applied to algebraic
topology and algebraic geometry and category theory - its insights are
not only too complicated, but the really useful ones usually come from
outside algebra where (Probable) Causation/Influence dwells.
For those interested in the dependence kernel measure literature, over
200 papers can be found at least somewhat related under the keyword
"feature" in Front for the Mathematics ArXiv, and a few dozen others
under "landscape," "(dependence) kernel," etc.
Osher Doctorow
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