# Re: Probabilities vs. grades of membership

Ellen Hisdal (ellen@ifi.uio.no)
Sun, 16 Aug 1998 04:58:28 +0200 (MET DST)

On Aug 9-th "Alp Atik" <atik@ptw.tu-darmstadt.de> writes:

> 1. Transforming probability distribution into fuzzy sets

> Is it possible to fulfill this transformation directly (how? any
> literature?) or is it necessary to transform them first into membership
> distributions (resp. possibility theory - dubois/prade) and in a second =
> step
> into fuzzy sets (e.g. civanlar/trussel).
> I have also some problems with the differences between the membership
> distribution and the fuzzy sets. Can someone explain me the differences
> conceretly? What is the decisive criterion?

See fig. 5 (including caption) of
@article{are,
author = {Hisdal, E.},
title = {Are Grades of Membership Probabilities?},
journal = {Fuzzy Sets and Systems},
year = {1988},
volume = {25},
pages = {325-348} }.

Loosely speaking, this paper shows that in contrast to the probability
P(u|tall) (of a given numerical height of objects labeled `tall'),
the grade of membership for the label `tall' can be
interpreted as the probability P(tall|u) of the assignment of the label
`tall' to an object of measured height u.

More details, as well as the detailed mathematical formula
for the connection between probabilities and grades of membership
can also be found on the internet, eqs. (21), (22) of

@techreport{inf1p4,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.4. {T}he {TEE} Model},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-054-1.
Can also be found on
http://www.ifi.uio.no/\$\sim\$ftp/publications/research-reports/Hisdal-4.ps},
number = {148} }

Greetings,
Ellen Hisdal

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