Is there any method to convert the fuzzy (possibility) distributions

with membership functions 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?

They are in general unrelated, so no.Quote:>Is there any method to convert the fuzzy (possibility) distributions

>with membership functions to probability distribution?

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.

---

Regards,

Dmitry Kazakov

www.dmitry-kazakov.de

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