| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
05 Jun 2005 10:05:07 AM |
| Object: |
Causation "Underlies" Spacetime? Part 2. |
From Osher Doctorow
COPYRIGHT NOTICE
Causation "Underlies" Spacetime? Part 2.
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
The solution to the problem of how Causation can underlie spacetime
appears to be a rather surprising one: re-interpret PI, Conditional
Probability (CP), and Independent Probability (IP).
Instead of these respectively being applicable only to Rare Events
(probability less than .05 Events), Fairly Frequent Events (probability
between .05 and .95), or Very Frequent Events (probability greater than
..95 Events) except at rather small intersections for the first two at
least, the scenarios appear to be quite different. All three of PI,
CP, and IP are applicable to Events other than Rare Events, while only
PI is applicable to Rare Events without distortion or "blowing up".
Each, however, gives a somewhat different aspect of "generalized
Causation" (GC). PI gives the Probability of Causation between Cause
and Effect and of Influence between them simultaneously. CP gives the
aspect of this from the viewpoint of the Effect with the Cause fixed or
given, which aspect is not a set/Event but as we know a ratio of
probabilities of set/Events except at denominator (probability of
Cause) 0. IP insofar as its Causation/Influence is concerned is split
into two types, IP1 and IP2, respectively defined by:
1) P(B|A) = P(B) when P(A) is not 0
2) P(AB) = P(A)P(B) for all P(A), P(B)
The two IP types coincide when P(A) is not 0 since P(B|A) is defined
as P(AB) divided by P(A) when P(A) is not 0. However, when P(A) is 0,
the second type still operates together with PI. Notice that
Statistical Independence even in the Mainstream literally is defined by
(2), sometimes called Mutual Independence for 2 or more sets (with
appropriate multiplications for the extra sets).
We know that:
3) P(A-->B) > = P(B|A) > = P(B) > = P(AB)
where the second and third inequalities hold only for positively
quadrant dependent Events (Events and via appropriate definitions
random variables whose probabilities or values increase together).
However, this is no longer regarded as a reason for splitting them into
mostly disjoint regions of probability space.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation "Underlies" Spacetime? Part 2. |
05 Jun 2005 10:44:02 AM |
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From Osher Doctorow
By not regarding the inequality (3) of last time as decisive in
splitting the probability space, the main result is that, barring
further discoveries at least, we simply compare values of PI, CP, and
IP respectively inside each of their classes. So we compare PI values
to other PI values, CP (Conditional Probability) values to other CP
values, etc.
Sometimes the results of calculating CP seem more intuitively
"reasonable" than the results of calculating PI or vice versa, but this
is not decisive either in either the physical or mathematical
interpretations.
For example, if we randomly toss two coins of the same type which are
not "loaded" (one is not heavier than the other, etc.), then we get:
1) P(B|A) = P(AB)/P(A) = P(A)P(B)/P(A) = P(B) = 1/2
which seems very intuitive since P(A) = P(B) = 1/2 where A is the Event
"lands with heads up" (heads for short), B is the Event "lands with
tails up" (tails for short).
The calculation of P(A-->B) yields:
2) P(A-->B) = 1 + P(AB) - P(A) = 1 + P(A)P(B) - P(A) = 1 + 1/4 - 1/2 =
3/4 = 0.75
This doesn't seem intuitive, but that's because we aren't familiar with
or used to comparing P(A-->B) values with each other. The relevant
comparison is not between P(A-->B) = 0.75 and P(B|A) = 1/2, but between
two different P(A-->B) values and two P(B|A) values. Since both are on
a scale of 0 to 1, the 0.75 value is 3/4 up its scale, and the 1/2 or
0.5 value is 1/2 up its scale, but that's all we can conclude in an
"inter-comparative" or "comparative" sense. There is nothing "magic"
about the number 1/2.
Shouldn't we intuitively believe that the two coins experiment would be
halfway up the PI scale in the sense that "independence" shouldn't be
more "highly causal" rather than "less causal"? As far as either the
mathematics or physics goes, there seems to be no basis for this
conclusion other than familiarity with the number 1/2 as a "compromise"
between 0 and 1.
But how can the causation between two Events be less than the causation
between two randomly tossed non-loaded coins?
In the same way that the CP (Conditional Probability) for two events
can be less than the CP for the two coins, PI can be less than 0.75.
There are much more "bizarre" events than two coins allegedly not
exerting causation on each other. For example, there are events which
sometimes do and sometimes don't seem to exert causation with no known
pattern. In the quantum world, for example, the behavior of the two
tossed coins would be (if it were microscopic or had a microscopic
analog) unusually stable "causation"! After all, they both land
similarly, they both are tossed similarly, and they both tend to have
probabilities of 1/2 in their outcomes. Since non-spurious (non
"coincidental") correlation is causation, on a correlation scale in a
"thought-experiment", 1/2 is pretty good! Of course, correlation
scales are another topic altogether (they're quite messy).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation "Underlies" Spacetime? Part 2. |
05 Jun 2005 11:07:03 AM |
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From Osher Doctorow
The real open question is this: what does the reversal of the
inequality (3) mean depending on whether two Events are positively or
negatively quadrant statistically dependent?
In other words, the respectively positively or negatively quadrant
statistically dependent cases of Lehmann in the 1960s are:
1) P(AB) > = P(A)P(B)
2) P(AB) < = P(A)P(B)
Lehmann formulated this in terms of random variables, but we can do
that by setting A = {w: X(w) < = x}, B = {w: Y(w) < = y} for two
continuous random variables X and Y for example, in which case (1) for
example becomes:
3) P(X < = x, Y < = y) > = P(X < = x)P(Y < = y)
where the comma , denotes "and".
So in other words, the real open question is why do the following two
"inequalities" respectively hold for positively versus negatively
quadrant dependent Events:
4) P(A-->B) > = P(B)
5) not necessarily P(A-->B) > = P(B)
Mathematically we can explain these as due to the definitions of
positive versus negative quadrant statistical dependence, but
physically is where the question is open. Events that are negatively
quadrant dependent (go probabilistically in opposite directions
regarding increase vs decrease in their values) mathematically reverse
the P(B|A) > = P(B) inequality and instead of reversing the P(A-->B) >
= P(B) inequality they do or don't depending on the Events! It would
be interesting to discover which types of Events do or don't reverse
the inequalities in the second case.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation "Underlies" Spacetime? Part 2. |
05 Jun 2005 02:01:19 PM |
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From Osher Doctorow
The mystery is solved, and is thrown back in the CP (conditional
probability) scenario.
The equation:
1) P(A-->B) > = P(B)
always holds! This is because:
2) 1 + P(AB) - P(A) > = P(B)
iff P(A) + P(B) - P(AB) < = 1, but the left hand side is P(A U B), and
P(A U B) is always less than or equal to 1 by definition of
probability!
Hence the only thing that reverses is P(B|A) < = P(B) versus P(B|A) > =
P(B), which are simply due to the shift from negative to positive
quadrant dependence.
Osher Doctorow
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