Hello,
I would like to comment a little bit on the relationship/differences
between fuzzy sets and Bayesian nets.
First of all, Bayesian networks is a model to reasoning under uncertainty.
Fuzzy sets theory is a model, extension of classical set theory, to
represent the fuzziness or vagueness inherent to gradual rather than crisp
(well-defined) notions or properties.
So, in principle the aims of both models are completely different and thus
they can be considered as orthogonal as Pearl suggest.
However fuzzy sets can be linked to a particular model of uncertainty
which is called Possibility theory, which again has nothing (or little) to
do with probability.
The idea is very simple. Suppose you are interested in the likelihood of
an event "the value of variable x belongs to a set A", where A is a
(classical) subset of the (discrete) domain of the variable x. For
instance, let x be "temperature of a certain device", A = [20, 30].
Consider the following three scenarios:
1) You are lucky and you are provided with a probability distribution p of
variable x. Then you can evaluate the likelihood of the event simply as
summing up the probability values p(t) for all t in A.
2) Much worst, you are only provided with the information that only values
of another subset B (e.g. [25, 35]) are feasible. In such case you can
only evaluate about the possibility of the event:
- "x in A" is possible event if A intersects B
- "x in A" is an impossible event otherwise
3) Still worst than 1. Suppose that you are told that only "high
temperatures" are feasible for that device. What to do in a such a
case?. Assume that "high
temperatures" can be represented as a fuzzy set with a membership function
m_high. Notice that m_high(t) is an evaluation to what degree a particular
temperature t can be considered high in the current context.
For instance we may have m-high(10) = 0, m_high(20) = 0.5, m_high(30) = 1,
m_high(40) = 1.
Now, POssibility theory postulates that the information "only high
temperatures are feasible" induces a "possibility distribution" on which
values the variable can take (this is uncertainty modelling!), and
moreover it postulates which is this distribution, it is just
poss(X=t) = m_high(t)
Notice that, if you know that "only high temperatures are feasible", to
have both poss(x= 30) = poss(x= 40) = 1 is perfectly reasonable.
A possibility distribution allows to evaluate how "possible (or
plausible)" and how "necessary (or certain)" is then any event:
Poss("x in A") = max{ poss(t) : t is in A}
Nec("x in A") = 1 - Poss( x is not in A) = inf{ 1-poss(t) : t is
not in A}
Thus the possibility degree of an event is determined by its highest
plausible elementary event.
If the fuzzy set "high temperatures" turns out to be defined as a
classical (non-fuzzy) set B, then we come down to the above sceneario 2.
In summary, Possibility theory is an (full flecthed) uncertainty model,
related to fuzzy set theory, different from probability theory, which
takes into account "weak" forms of information and thus with a qualitative
flavour and can be used if "strong" numerical information is missing.
Hope this might help you.
There is plenty of (serious) bibliography about all this stuff if you are
interested.
Lluis Godo.
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