## Convert Fuzzy distributions to Probability distribution

### Convert Fuzzy distributions to Probability distribution

Is there any method to convert the fuzzy (possibility) distributions
with membership functions to probability distribution?

### Convert Fuzzy distributions to Probability distribution

As far as I know it is not possible to convert fuzzy membership function
to probability distribution.

Probability theory and fuzzy sets theory are working
on different kinds of uncertainty.

In probability theory the basic problem
is how often something happened.

In fuzzy set theory people are tiring
to guess for example what to do with Robert who is
half-German, quarter- French and quarter- Italian.
The numbers 0.5, 0.25 and 0.25
are not probabilities but grades of membership or, equivalently truth
values.

I collect different points of view
about relations between probability and possibility
on the following page:

http://zeus.polsl.gliwice.pl/~pownuk/fuzzy.htm

If you will get a headache after reading different opinion about that,
then don't worry.
This topic is still under investigation.

Regards,

Andrzej Pownuk
http://zeus.polsl.gliwice.pl/~pownuk/

Quote:> Is there any method to convert the fuzzy (possibility) distributions
> with membership functions to probability distribution?

### Convert Fuzzy distributions to Probability distribution

Quote:>Is there any method to convert the fuzzy (possibility) distributions
>with membership functions to probability distribution?

They are in general unrelated, so no.

However, in particular cases there could be a relation. Very often a
physical measure, which is by its nature random is replaced by some
fuzzy thing. This discards a lot of useful information, but also
allows us to deal with complex data in an easier way. A typical
example is some measurement data given in the form of intervals, like
t=[12.5,15.5].

Dubois and Prade in various works describe an approach to fomalize
relations between possibility and probability for these cases. In
short they go as follows. Let there is a distribution of
probabilities. Let's build a set of intervals (better around the mean)
so that each new interval contains all others. It is then easy to see
that if we consider subsets of the set of these intervals, then the
probabilites will obey min-max axioms of the fuzzy sets, These
intervals Dubois and Prade call "focal elements". For focal elements
they postulate Pos(Fi)=Pr(Fi) (in crisp case, in fuzzy case, Nec comes
into paly). So here you are.

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Regards,
Dmitry Kazakov
www.dmitry-kazakov.de