Re: Fuzzy MLE and Fisher Information Analogue?

Stephen Paul King (stephenk1@home.com)
Tue, 11 May 1999 05:02:38 +0200 (MET DST)

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:
http://www.arc.unm.edu/Conferences/Roy_Frieden_Abstract.html

> 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,

Stephen

> Cordially, phil diamond

stephenk1@home.com (Stephen Paul King) wrote:
>Hi all,
>
>> Subject:
>> Re: Fisher information
>> Date:
>> Wed, 5 May 1999 10:53:19 -0400
>> From:
>> "Christopher Brown" <cbrown@chem1.chem.dal.ca>
>> To:
>> <stephenk1@home.com>
>
>>> I have assembled a link page on Fisher information and have a
>>>definition: "The Fisher Information about a parameter is defined to
>>>be \theta the expectation of the second derivative of the
>>>loglikelihood."
>>>http://members.home.net/stephenk1/Outlaw/fisherinfo.html
>>> But I am still needing an intuitive grasp of that it means. :)
>
>
>>In short, when you estimate a parameter, you estimate it's value usually by taking
>>the estimate of the parameter to be the maximum likelihood value. So we get an
>>estimated parameter value, and we know it's uncertain. Imagine it as a normal
>>distribution, the center of which is our estimate, and the variance of which is the
>>uncertainty we have in the location of our estimate. The Fischer Information
>>essentially describes how sharp that normal distribution is around our estimate.
>>More Fischer Information roughly implies a more informative estimate (i.e. tighter
>>spread around the MLE).
>
>>Hope it helps,
>>CDB
>
> 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?
>
>Kindest regards,
>
>Stephen

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