[time 286] Re: degeneracy of distributivity

Stephen P. King (stephenk1@home.com)
Fri, 07 May 1999 10:16:44 -0400

Dear Phil,

Phil Diamond wrote:
> In comp.ai.fuzzy you write:
> > Is there a fuzzy logical version of this, e.g. the MLE and the
> >consequent definition of the Fisher Information? Also, is there a way
> >to parametrize the sharpness of the fuzzy normal distribution, perhaps
> >as a function of the \theta parameter?
> I have only come into this thread now and haven't read earlier posts.
> However, there are significant problems in even defining the normal
> distribution for fuzzy random variables. This is related to the same
> difficulty with convex compact valued random variables (NN Lyashenko,
> J. Soviet Math 21 (1983), 76-92). The only normal distribution is
> degenerate in the sense that it is a constant set (the expectation)
> translated by a normally distributed random vector. Since the level
> sets of FRV will be convex, compact sets in the mathematically
> tractable cases, the same degeneracy arises. However, since I have
> not read what others wrote, this may be entirely different from what
> you are talking about.

        I think that it is what I am talking about! :) This degeneracy of
distributivity is what I thinking about. ;) I am thinking that the
dificulty may be dealt with in a practical sense by showing that
computations of asymptotic approximations to a norm (which is some sort
of limit -> oo) are possible for some classes of finite state systems
(involving bisimulation).
        Has any study been done on the nature of this "normally distributed
vector"? Is is complex in the Chiatin sense?
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/max.html (equating the
vector to a bit string)
        I also found: http://www.mathe.tu-freiberg.de/math/publ/pre/95_12/
which is apparently (from the abstract) dealing with this question! :)
        I have been trying to make sense of B. Kosko's ideas of a "information
wave equation": in chapter VII of his book Fuzzy Engineering
[http://www.prenhall.com/books/esm_0131249916.html] and comparing it to
B. Roy Frieden's resent work:

> I have a paper about MLE for fuzzy linear models based on uniformly
> distributed errors (in Springer Lecture Notes in Computer Science
> No. 313, 1988). Even here, the results are pathological and the
> estimators are second order fuzzy sets. You can find out a little
> more on fuzzy random variables in the paper "Fuzzy Kriging", FSS
> 33 (1989), 315-332. A very terse description is given in Diamond
> & Kloeden, "Metric Spaces of Fuzzy Sets", World Scientific, 1994.
[e.g. (http://wspc.com.sg/books/mathematics/2326.html)]

        Would you happen to have poscript or TeX e-versions of these papers? I
will try to order the book...

Kindest regards,

> Cordially, phil diamond

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