From: mailbox@dmitry-kazakov.de   
      
   On 20 Aug 2003 07:58:46 -0700, aalwazeer@gistrans.com wrote:   
      
   >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.   
      
   ---   
   Regards,   
   Dmitry Kazakov   
   www.dmitry-kazakov.de   
      
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