Re: Phases Underlie Probability-Statistics: PI, FFE, VFE.

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 12/24/04 

Date: Fri, 24 Dec 2004 13:41:44 +0000 (UTC)

On 23 Dec 04 11:35:49 -0500 (EST), Osher Doctorow wrote:
>The current indications are that Knowledge and Information are
>encoded in the geometry of spacetime in such a way that (using
>"Information" language although I am more interested in Knowledge
>as "Semantics" rather than syntactics) they are not lost when
>objects enter black holes. They are encoded in the near-black-hole
>geometry outside black holes.
>How is this possible? Probable Influence (PI) suggests that we
>look for an answer in the "frontier" between the microscopic and
>macroscopic worlds. Recall that the reversibility of PI P(A1-->A2

At this point, we are beginning to get the vague suspicion that
PI may be encoded in geometry not only near black holes but else-
where including in human memory at least. But isn't that only a
small part of the story of cognition and of propositions? Surely
the Piron-Jauch "yes-no" propositions describing quantum experiment
results only turned out to be part of the whole quantum and macro-
scopic worlds? Surely there is more to human thought and experience
than logic or modified logic?

Yes and no, to be somewhat facetious but with a bit of reality thrown
in. Logic has already won the quality vs quantity argument in favor
of quality, being qualitatively different from most other things and
yet having some quantitative measures in their set/event analogs
(probability being such a measure). But in a partial way, logic can
also win an incredible argument or dispute: the dispute over Know-
ledge at least as semantic "information". It does this for Rare
Events by constraining other types of Knowledge as follows.

Let us look again at "information", let us say at the Shannon type
of information, which "comes into its own" with mutual information
and conditional information, the former an intersection type of
information of two events, the latter a "given" type of information.
I claim that it is already too complicated - that Shannon informa-
tion and entropy "overkills", and by similar arguments so do
various related types of information/entropy.

Theorem. For all sets/events A, B, we have:

1) P(AB) < = P(A)P(A-->B)
2) 0 < = P(A)P(A-->B) - P(AB) < = min{P(A), P(A-->B)}

Proof. Since P(A-->B) = 1 + P(AB) - P(A), (1) holds iff:

3) P(AB) < = P(A)[1 + P(AB) - P(A)] = P(A)(1 - P(A)) + P(AB)P(A)

which holds iff:

4) P(AB)[1 - P(A)] < = P(A)[1 - P(A)]

If P(A) = 1, (4) holds. If P(A) is not 1, (4) holds iff P(AB) < =
P(A) which always holds by monotonicity of probability. As for (2),
the first inequality of (2) is the same as (1), and the second
inequality of (2) follows from P(A)P(A-->B) - P(AB) < = P(A)P(A-->B)
< = P(A) because P(A-->B) < = 1 and P(A)P(A-->B) - P(AB) < = P(A-->
B) since P(A) < = 1. Q.E.D.

Remark. It follows from the Theorem that when P(A) is small (Rare
Events), so is P(AB), and that when P(AB) is relatively large, so
is P(A-->B). Since we want to keep P(A) and P(AB) small for Rare
Events, the point of the second condition here is that as P(AB)
increases for Rare Events, P(A-->B) is forced to have a higher
greatest lower bound (glb). In addition, if P(A-->B) gets small,
then P(AB) gets closer to P(A)P(A-->B) from the Theorem. These are
quite a few constraints on Knowledge of P(A) and P(AB). Admittedly
they do not completely determine P(A) and P(AB), and in my opinion
Knowledge is not completely determined quantitatively, but it is
at least as good as Shannon and similar modified/generalized types
of information/entropy can do and with far fewer restrictions as to
types of messages, channels, speeds, etc.

Osher Doctorow

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