Science > Physics > Optimal vs Suboptimal Probable Influence/Causation
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
24 Jul 2005 01:57:25 AM |
| Object: |
Optimal vs Suboptimal Probable Influence/Causation |
From Osher Doctorow
COPYRIGHT NOTICE
Optimal vs Suboptimal Probable Influence/Causation
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
The reason why correlation does not imply causation (but causation
implies correlation in mathematical probability-statistics!) is that
correlation may be "spurious", which is to say coincidental since
unrelated objects may show similar behavior over a certain period.
Probable Influence/Causation (PI) defines a non-spurious correlation
between A and B as:
1) Non-Spurious Correlation P(A,B) = P(A<-->B)
This generalizes to n set/events A1, A2, ..., An by:
2) P(A1,A2,...,An) = P(A1<-->A2<-->...<--> An), n = 1, 2, 3,...
Unlike typical Mainstream mathematical probability-statistics
correlations which involve averaging via expectations or expected
values, P(A,B) and its generalizations are defined pointwise when
random variables X, Y are involved, with A = {w: X(w) < = x}, B = {w:
Y(w) < = y} so that P(X,Y)(x,y) is the correct notation for two random
variable P(A,B), and it changes from point to point.
I'll use NSPC as "Non-Spurious Probable Correlation" for short as
defined by (1) and (2), and it is immediate from the definition that:
Theorem 1. NSPC does not depend on the order in which events or random
variables probably influence or cause each other and is thus optimal in
this sense, while it is optimal in value if it equals 1.
Are there any suboptimal results? Yes. They are based on the
following types of theorems.
Theorem 2. P(A-->B) - P(B-->A) = P(B) - P(A)
Theorem 3. P(A-->B-->C) - P(B-->A-->C) = P(A'C) - P(B'C)
Theorem 4. P(A-->B-->C) - P(B-->A-->C) = P(C-->B) - P(C-->A)
The proofs are rather simple, noting that (A-->B-->C) = (A-->B)(B-->C)
where adjacent parentheses are intersected, and (A-->B) = A' U B, so
that (A-->B)(B-->C) = (A'UB)(B'UC) = A'B' U A'C U BC
because BB' = N (the null set). I'll leave them as an exercise,
although I may supply them later.
Theorem 2 says if you interchange the order in which two events
influence each other, then the probable influence of the original
ordering exceeds the probable influence of the changed ordering to the
extent that the probability of the originally influenced event exceeds
the probability of the originally influencing event.
Theorem 4 is similar to theorem 3 except that "originally influenced
event" is now the last influenced event in a chain of three events and
"originally influencing event" is now the first in a chain of three
events, and "probability of the originally influenced/influencing
events" is replaced by the probable influence between the two extreme
events mentioned.
Theorem 3 says roughly that for interchanging the order of two out of
three events to make a big difference, the last influenced event of
each ordering should differentially probabilistically intersect the
outside (complement) of the first influenced event in the order
indicated. A similar "differential probability" (in the sense of very
different probability) characterizes theorem 4 with the intersections
replaced by two-event influences.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Optimal vs Suboptimal Probable Influence/Causation |
24 Jul 2005 02:10:10 AM |
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From Osher Doctorow
What is suboptimal is one-sided influence, of form P(A-->B) or
P(A-->B-->C) or in general P(A1-->A2-->...-->An). In other words, the
theorems stated enable one to obtain bigger or maximal one-sided
probable influence, usually recursively in a sense via smaller or fewer
sets. Numerically suboptimal one-sided influence is still 1 in value,
although one can of course settle for 1 - epsilon for small positive
epsilon probable influences.
Notice from theorem 1 that if A and B are in the same orbit in the
sense that P(A) = P(B), then interchanging their order makes no
difference in terms of probable influence. Thus, events having the
same probability anywhere in the universe have resilience or robustness
with regard to the order in which they occur at least in a
probabilistic suboptimal sense! This is advantageous in terms of
optimality but not suboptimality since one event will not gain
probabilistic advantage over the other via changing their order.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Optimal vs Suboptimal Probable Influence/Causation |
24 Jul 2005 02:23:51 AM |
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From Osher Doctorow
We can actually calculate the difference between optimality and
suboptimality, which I will do for three events below.
Theorem. P(A-->B-->C) = P(A<-->B<-->C) + P(A' C)
The proof is simple and will be left as an exercise. Notice that
one-sided probable influence for 3 events is in general at least as
probable as two-sided probable influence (probable correlation) for the
3 events, and the discrepancy increases to the degree that P(A' C)
increases, which is to say that the probability that the last
influenced event intersects the outside of the first influencing event
increases. If the last influenced event doesn't intersect the outside
(complement) of the first influencing event, then suboptimality equals
optimality and probable correlation which is P(A<-->B<-->C) is what
P(A-->B-->C) reduces to.
A similar theorem for two events A, B can be proven, namely P(A-->B) =
P(A<-->B) + P(BA' ).
Osher Doctorow
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