Below are some paragraphs of a (LATEX) paper which I have just written for
the FLINS 96 volume. They concern your subject.
Best greetings,
Ellen Hisdal
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\section%
{Two Aspects of the Probability-Possibility Discussion\\
and the Probabilistic Fuzzy-Settians}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Two Aspects of the Discussion}
The discussion of probabilities versus possibilities has two different
aspects.
On the one hand we have those who are opposed to fuzzy set theory;
and who assume that probabilities and possibilities are rivals,
both of which claim to describe the same concept.
They then show, or try to show, by examples that the probabilistic
operations give better results than Zadeh's max and min operations.
The papers of references \cite{lindley}, \cite{ls} and \cite{woodall}
belong to this group.
On the other hand, the fuzzy-settians believe that probabilities and
possibilities denote completely different concepts.
However, the probability-minded subset of this group,
we will call them the {\em probabilistic fuzzy settians}, believe that
possibilities {\em can} be interpreted with the aid of the theory of
probability. The papers of references ...
belong to this group. Furthermore I believe that Bandler and Kohout's
checklist paradigm \cite{bkchecklist} uses the probabilistic
fuzzy-settian interpretation of
grades of membership, although they do not say so explicitly.
The papers by Bandler and Kohout, Mabuchi and Beliakov deserve special
attention because they treat the difficult multiattributional case.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Probabilistic Fuzzy-Settians}
The probabilistic fuzzy-settians try to clarify
the {\em meaning}
of grades of membership or possibilities%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\foot{A careful reading of Zadeh's main paper on possibilities
\cite{zposs} reveals that possibilities and grades of membership
have always the same numerical value for the same $(\lambda,u)$ pair.
Consequently they must denote the same concept. We will therefore also use
possibilities and grades of membership as being different words for
the same thing.}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
They are able to show that their interpretation of
possibilities results in more generally valid
formulas for the connectives than the
max-min interpretation. Furthermore they can {\em derive} the formulas
for the connectives instead of having to postulate them.
The formulas for the connectives according to the TEE model
are {\em given} in \cite{are}. They are {\em derived} in
\cite[sect.\,10]{64}.
The `one-minus'
formula for the negation follows directly from the probabilistic
fuzzy-settian interpretation of possibilities.
Probabilities of \lambda\ (e.g. \lambda=tall) and possibilities of \lambda\
are both functions of the pair of values $(\lambda,\,\ux)$ (e.g.
(tall,\,165cm)~), where \lambda$\in$\Lambda\ is an element of a `complete and
nonredundant label set' $\Lambda=\{\lambda_1,\ldots,\lambda_L\}$,
for example $\Lambda=\{\mbox{short, medium, tall}\}$
(see footnote~3).
$\ux\!\in\! \Ux$ is the measured attribute value of the object.
Superficially expressed, the difference between
$P_{\lambda}(\ux)$, the probability distribution of \lambda,\ \ and
$\Pi_{\lambda}(\ux)$, the possibility distribution of \lambda,\ \ is that
$P_{\lambda}(\ux)$,
is equal to $P(\ux|\lambda)$, the conditional probability
that an object labeled~\lambda\ has the attribute value~$\ux$.
While the possibility $\Pi_{\lambda}(\ux)= \mu_{\lambda}(\ux)$
is identified with $P(\lambda|\ux)$.
This is the conditional probability
that an object with attribute value~$\ux$ will be assigned the label
\lambda$\in$\Lambda.
Probability and possibility distributions of \lambda\ are thus both
probability distributions, but over different spaces. The former is a
probability distribution over the space $\Ux= \{\ux\}$. The latter is a
probability distribution over $\Lambda=\{\lambda_1,\ldots,\lambda_L\}$ for a
given attribute value \ux.
Considered as a function of $\ux$,\ \
possibilities $\Pi_{\lambda}(\ux)= P(\lambda|\ux)$ are called likelihood
distributions of \lambda\ over $\ux$ in the theory of probability
(see \cite{fisherbox} for an interesting historical discussion
of likelihoods and prior probabilities).
The summing-up-to-1 law of probabilities thus takes the form,\vspace*{-1ex}
\beq
\sum_{i=1}^I\,P_{\lambda_l}(u_i)=
\sum_{i=1}^I\,P(u_i|\lambda_l)=1
\qquad
\sum_{l=1}^L\,\pi_{\lambda_l}(u_i)=
\sum_{l=1}^L\,\mu_{\lambda_l}(u_i)=
\sum_{l=1}^L\,P(\lambda_l|u_i)=1\vspace*{-0.8ex}
\eeq
for probabilities and possibilities respectively.
When the possibilities refer to a yes-no question then the right hand
sums have only the two terms $l= 1=$yes-\lambda, and
$l= 2=$no-\lambda. {\small (For the formula connecting probabilities and
possibilities, see \cite{inf1p4}, eqs.\,(18).\,(19) and fig.\,4.)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Bezdek's Bottles}
The following example is given by Bezdek \cite[p.\,43]{bottles}
to show that
``fuzzy sets are not just a clever disguise for statistical models.''
A thirsty traveler has been in the desert for a week without drink
and comes upon two bottles marked A and B.
Furthermore bottle~A is marked with `$\mu_{potable}= .91$', and
bottle~B is marked with `$P_{potable}= .91$'.
The universe of objects is the set of all liquids according to Bezdek.
The fuzzy subset~L of this universe is defined as the set
of all potable liquids (potable=suitable for drinking).
The traveler must now decide which bottle to drink from.
Bezdek now claims that the traveler should choose bottle A
which may contain, for example, swamp water [or beer]; indicating that
the contents of~A are ``~`fairly similar' to perfectly potable liquids
(pure water)''. While bottle~B has a 91\% chance of containing poison.
According to Bezdek this example illustrates the different kinds of
information conveyed by fuzzy memberships versus probabilities.
I think that all of us can agree with Bezdek this far. However, this
does not mean that the theory of probability cannot be used to explain
what \ms s are about.
Let us start with bottle~B. Its probability value indicates that it
was, chosen randomly from a store of, e.g., 1000~bottles
about which it was known that 910 contain pure water and 90 contain poison.
To interpret the {\em \ms} value 0.91 we choose a completely different
probability space. Namely the space consisting of the two labels
$\{\mbox{potable, \NOT\ potable}\}$.
The person who glued the label `$\mu_{potable}= .91$' on~A could
have taken into account that 9\% of all people (by her
own estimate), e.g.
teetotalers and believing moslems, consider beer to be `\NOT\ potable';
while the other 91\% consider it to be `potable'.
This is a typical case of taking
fuzziness \#3 (see \fig{figsourcesoffuz}) into account;
namely the possibility that different people may have
different ideas concerning the appropriateness of a given
label to an object.
We see that the objection to a probabilistic interpretation of grades
of membership holds only when the same probability spaces are used for
$P$ and $\mu$. Everything falls into place when the probability space
of \ms s is a legal label set (see footnote~3).