Science > Physics > Invariance As Part of Intersection Theory 4: P(A<-->B) = P(AB) + P(A' B' )
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
04 Sep 2005 07:45:34 PM |
| Object: |
Invariance As Part of Intersection Theory 4: P(A<-->B) = P(AB) + P(A' B' ) |
From Osher Doctorow
The Probable Correlation Theorem is:
1) P(A<-->B) = P(AB) + P(A'B' )
where A<-->B is defined as:
2) (A-->B) = (AB' )' = A' U B
The expression (A<-->B) is the set theory analog of logical "a iff b"
where a, b are propositions (for example, they could be respectively
taken as the propositions that events A, B hold). It expresses "A and
B mutually influence or cause each other". This is the essence of
correlation except that non-spurious correlation does not enter the
mathematics.
When A or A(x) = {w: X(w) < = x}, B = {w: Y(w) < = y} for two
continuous random variables X, Y, where B can also be written B(y), for
x and y real, then P(A<-->B) becomes F(x,y) + P(X > x, Y > y) where the
comma indicates "and", and F(x,y) is the joint cumulative distribution
function (cdf) of X and Y at (x,y). In that case, we write P(A<-->B)
as P(X<-->Y)(x,y) or drop (x,y) if the argument is understood:
3) P(X<-->Y)(x,y) = F(x,y) + P(X > x, Y > y)
This Probable Correlation remains non-spurious as you can see from the
right hand side of (3), and it is calculated pointwise at each point
(x,y) rather than averaged as an aggregate. So no information or
knowledge is lost, unlike the usual correlation in mathematical
probability-statistics which is averaged.
It is accurate to call the left hand side of (3) the Probable Mutual
Influence or Mutual Causation of X and Y, although Probable Correlation
is also correct if spurious correlation is understood to not enter it.
Readers will notice that the right hand side of (3) and the right hand
side of (1) are decompositions into two intersections or their
measures/probabilities. So Probable Correlation is completely based on
intersections and a decomposition into intersections.
Osher Doctorow
.
|
|
| User: "OsherD" |
|
| Title: Re: Invariance As Part of Intersection Theory 4: P(A<-->B) = P(AB) + P(A' B' ) |
04 Sep 2005 08:19:53 PM |
|
|
From Osher Doctorow
Surely conditional probability, the only rival to Probable Influence
(PI), has some similar expression? No.
It is true that for one-sided Probable Influence P(A-->B), which
equals:
1) P(A-->B) = 1 + P(AB) - P(A)
there is a conditional probability "analog" P(B|A) which isn't
interpreted in the same way, namely:
2) P(B|A) = P(AB)/P(A)
where the slash on the right hand side indicates division and P(A) is
not 0. If P(A) is 0, then P(B|A) isn't defined. Notice that the two
factors P(AB) and P(A) of the right hand side of (2) are the two terms
of the right hand side of P(A-->B), but with division replacing
subtraction (plus 1).
P(B|A) is interpreted as the probability of B given either that A has
occurred or that we have information about A, although since no
"information apparatus" is defined, the interpretation "given that A
has occurred" is the more usual one in practice, abbreviated "given A".
But the word "given" can't quite be left as a "primitive" - it
literally means that A is fixed, and a similar usage occurs in Fubini's
Theorem in multiple/iterated integrals in real analysis where the inner
integral is fixed - but only as a stage in deriving something else. It
corresponds to the Radon derivative in real analysis.
Notice that there is something unfortunate about the definition of
P(B|A) in (2), namely that it is not the probability of a set but
rather the ratio of two probabilities (provided the second one isn't
0). In addition to the complication introduced by division by 0 (which
prevents super-rare events from being analyzed by conditional
probability, which also blows up in a small neighborhood of P(A) = 0)
the complication that P(B|A) is not a probability of a set means that
it cannot be generalized to a "mutual given" or "mutual influence" set
since the expression (B|A) is not a set, unlike the expression in
P(A-->B), namely (A-->B) = (A' U B). So there is no "Conditional
Probable Correlation".
There is a separately defined correlation coefficient in mainstream
probability-statistics defined as:
3) E[(X - EX)(Y - EY)]/[o(X)o(Y)]
where o(X) is the standard deviation of X (square root of its variance
E(X - EX)^2 ). It is symbolized rho(X, Y). It aggregates or averages
and is still a quotient and is not derived from generalizing or even
integrating P(B|A) or its "argument". The rough intuitive idea is that
if X is much greater than (or much less than) its population mean or
expectation EX and likewise for Y, then rho(X, Y) will be larger in
absolute value, while if X is near its mean and Y is near its mean,
then rho(X, Y) will be near 0. The denominator is always positive.
Osher Doctorow
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Invariance As Part of Intersection Theory 4: P(A<-->B) = P(AB) + P(A' B' ) |
04 Sep 2005 08:28:44 PM |
|
|
From Osher Doctorow
I meant to type (2nd posting back):
1) P(A<-->B) = P{(A-->B)(B-->A)} = P{(A' U B)(B' U A)}
where (A-->B) is defined as:
2) (A-->B) = (AB ' )' = A' U B
and where:
3) P(A-->B) = P(A' U B) = 1 + P(AB) - P(A)
The expression (A-->B)(B-->A) is the intersection of the sets (A-->B)
and (B-->A).
Osher Doctorow
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Invariance As Part of Intersection Theory 4: P(A<-->B) = P(AB) + P(A' B' ) |
04 Sep 2005 09:07:30 PM |
|
|
From Osher Doctorow
We have then only one real "rival" in physics applications, the
Heisenberg Uncertainty Principle (HUP), which dispenses with the
intersection of measuring position and momentum and claims that if the
variance or standard deviation or "uncertainty" of one is small, then
the corresponding aggregated or averaged quantity of the other is big
or vice versa (up to a very small specified constant).
Do the 14 subfields or subdivisions of Invariance-Intersection Theory
tend to make HUP or PI intuitively plausible? Of course, there's no
question about it - they tend to make PI intuitively plausible and to
mitigate against accepting any claim about non-existing intersections.
But in the main places where HUP has been applied, in quantum
fluctuations or foam and entanglement, is there not a duality or
opposite behavior of position and momentum based on both theory and
experiment? Yes, there tends to be a duality or opposite behavior.
But to leap from that to the conclusion that intersections are
undefined is unjustified. The most elementary "opposite behaviors"
have the forms:
1) x = k/p (p not 0, k constant)
2) x = k - p (k constant)
in the specific case of position (x) and momentum (p). But they don't
imply anything about nonexistent or undefined intersections. This is
what I have maintained all along in various threads. But I don't think
that I ever stated it as clearly and in such a clear context as in this
thread.
Isn't the double slit interference experiment an indication that HUP is
correct, and that "quantum strangeness" is a different kind of animal?
No, unless by "different kind of animal" we mean "involves more than
one dimension". We already knew that wave-particle duality, or
nowadays field-particle or particle-field duality, was plausible,
although non-elementary-particle physicists tend to keep the "particle"
part implicit because they don't quite know how to fit it into their
ideas. The simplest explanation of the double slit experiment is that
the moving/source or expanding/source object has two dimensions instead
of one: a mass or particle inertial dimensions, and a wave or field
dimension. A particle would go through one slit, a wave/field would
"flow" through both and arguably "know" what was later and before by
expanding/flowing. Depending on the experimental arrangement, the
wave-particle or particle-field would shift its behavior dimensionally.
It "sees" both dimensions simultaneously just as John at origin of
axes x, y, z, "sees" in the x, y, and z (positive or negative)
directions simultaneously.
Can mass as inertia and fields be different real dimensions just like
the 3 + 1 or 4 + 1 or 10-11 or 24 dimensions of classical or
Kaluza-Klein or superstring/brane physics? I have described the
plausibility of such scenarios in quite a few threads in the last year
on sci.physics. Mass arguably has 3 dimensions: mass as inertia,
energy, curvature. Time arguably has 3 dimensions: time as duration,
time as causation (since causes precede effects), time as transfer of
causation (transmission of causal knowledge or semantic information).
Force has 1 or 2 dimensions, either attractive or repulsive or
considering both to be part of the same dimension of force. Length has
3 dimensions (the usual x, y, z or Lx, Ly, Lz of dimensional analysis).
So mass and wave/field have no difficulty in being conceptualized as
having two simultaneous dimensions.
In the case of quantum entanglement, an appropriate laser beam on an
appropriate object yields twin "entangled" objects going off in
different directions, which can be made opposite or just different in
various directions. The entanglement lasts apparently indefinitely
with distance. We don't think that the "particles" are entangled since
particles or even little strings are usually regarded as having nothing
to entangle with. But their associated waves/fields can easily be
regarded as being entangled. They intersect! They intersect and
expand. So we've created an intersecting and expanding wave/field.
Notice that we haven't even violated locality, since the first
particle-wave or field-particle still retains itself and its wave
except that its wave is now expanding. In fact, there is nothing that
is not occurring locally in each wave-particle or particle-field. Two
such objects have to be there just as two people can't talk with each
other if they're not both somewhere, but there's no event or process
going on which does not reduce to any one object acting or operating
(though the action or operation may alternate from one to the other).
Osher Doctorow
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Invariance As Part of Intersection Theory 4: P(A<-->B) = P(AB) + P(A' B' ) |
05 Sep 2005 12:06:49 AM |
|
|
From Osher Doctorow
Now look at this Theorem:
Theorem. For n > = 1 integer, the Probable Correlation of n sets A1,
A2, ..., An is:
1) P(A1<-->A2<--> ... <-->An) = P(A1A2A3...An) + P(A1' A2' A3' ... An'
)
Proof. By mathematical induction:
2) (A1<-->A2<--->...<--->An) = (A1A2...An) U (A1' A2' ... An' )
where the two sets on the right hand side of (2) are disjoint (mutually
exclusive). So the probability of the left hand side of (2) is the sum
of the probabilities of its right hand side. Q.E.D.
Thus, unlike mainstream mathematical probability-statistics, PI
Probable Correlation generalizes to n sets and n random variables.
There is such a thing in mainstream probability-statistics as a
"multiple correlation coefficient", often denoted by R or R with
subscripts from 1 to n, R_(1...n), or similar notation, but it doesn't
generalize anything other than itself. It comes from statistical
multiple regression and as usual is not based on generalizing from one
to n sets. It has most of the defects of the usual mainstream
correlation coefficients and also of conditional probability.
Osher Doctorow
.
|
|
|
|
|
|
|
|
Related Articles |
|
|
