Re: [UAI] Fuzzy sets vs. Bayesian Network

From: Lluis Godo (godo@iiia.csic.es)
Date: Mon Feb 28 2000 - 11:49:03 PST

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    Hi all,
    I would like to contribute to this interesting discussion with some
    comments on Kathy's and Benjamin's opinions.

    Kathy:

    >I have heard exactly one proposed ontology for fuzzy membership functions
    >(proposed by Judea Pearl, among others) that makes sense to me. Under
    this
    >proposed ontology, the fuzzy membership of Robbie's adult height in the
    set
    >"tall," in a given context, should be taken as proportional to the
    >probability that a generic person in that context would use the label
    >"tall" to describe Robbie. Thus, fuzzy memberships are likelihood
    >functions. We can think of them as soft evidence applied to numerical
    >crisp set height measurements.

    This "soft evidence" is actually what possibility theory tries to capture
    and handle and I think there is some work done in relating likelihood
    functions and possibility distributions.

    There is another ontology, pushed by Enrique Ruspini (SRI), for fuzzy
    membership functions which I personally like, it is to interpret
    membership degrees in terms of similarity in the following terms. Using
    your example, the fuzzy membership of Robbie's adult height in the set
    "tall," in a given context, should be taken as proportional to how similar
    (in terms of height) Robbie is (or will be) to some pre-determined
    prototypical "tall" people in the given context. However, this ontology
    does not give rise to a fully truth-functional framework.
     
    >I might go beyond this and suggest an alternate criterion, that the fuzzy
    >membership be proportional to the *utility* for an appropriately defined
    >decision maker in that context of using the term "tall" to describe
    Robbie
    >(this, for example, would allow us to weigh costs of inappropriate usage
    of
    >the term).
    >
    >This proposed ontology makes a lot of sense at a surface level. However,
    I
    >know it is not what most fuzzy set researchers think they are talking
    about
    >when they use fuzzy memberships. I've never seen its mathematical
    >implications worked out, or seen any discussions about whether or under
    >what circumstances it gives rise to combination rules that look anything
    >like what the fuzzy people now use.

    I agree that one thing is to provide meaningful ontologies and another is
    to try to fit them with a bunch of combination rules proposed for fuzzy
    sets, some of them clearly not justifiable. This is a harder problem,
    although there is a growing interest in the fuzzy community to constrain
    arbitariness and at the same time to justify reasoning models on ground
    logical bases. This may lead hopefully to a convergent process.

    Benjamin:

    >Wrt fuzzy logic overall:
    > I always have found problematic the justification of the min or product
    >rules for the conjunction-combination operation in fuzzy logic, for a
    >couple reasons:
    >- each makes a very strong and rigid assumption about dependency
    (complete
    >dependence for min, complete independence for product), when viewed in
    >Bayesian terms.
    >- there is no sensitivity to what propositions are being conjunctively
    >combined, e.g., it could be P has membership of 0.4, not-P has membership
    >0.6, and then the logical contradiction (P and not-P) is assigned
    non-zero
    >membership: 0.4 (min rule) or 0.24 (product rule).
    >Boolean tautologies are thus not respected.

    I think here there is clear misunderstanding. Using Kathy's terminology,
    fuzzy logic is only appropriate for sets NOT satisfying the "clarity
    test."
    Therefore a "crisp" proposition (i.e. satisfying the "clarity test") even
    in a fuzzy logic setting has to be constrained to only take the Boolean
    truth-values 0 or 1. So, for crisp propositions, Boolean tautologies are
    preserved, not for fuzzy propositions. This can be readily seen by looking
    at the main systems of propositional fuzzy (many-valued) logic: Goedel
    logic, Lukasiewicz logic and Product logic. If you add to any of these
    systems the axiom
            A v not-A
    where v is max-disjunction, then all those systems collapse to Boolean
    classical logic. I might recommend the book "Meta-mathematics of fuzzy
    logic" by Peter Hajek (Kluwer, 98), to my mind the best manuscript about
    fuzzy logic from the strict and formal logical point of view.

    >
    >Wrt applications of fuzzy logic to control (where they have had a lot of
    >practical success and popularity in the last decade or so):
    >Here fuzziness is applied to beliefs about what actions to take.
    >I have yet to see a terribly satisfactory or deep intuitive
    >*Bayesian*-flavored account of fuzziness in this context, esp. of fuzzy
    >logic in the ways it is usually applied to control applications. One way
    >to think about this at a high-level would be to view fuzzy beliefs about
    >the control variables in terms of an oracle/person who would say what is
    >the probability, or expected utility, of the right thing to do, given
    >various evidences. But needed is a detailed Bayesian interpretation of
    >the kind of fuzzy logic rules that are typically found in successful
    >practice, in relation to this high-level interpretation.

    To my opinion the best way of interpreting what fuzzy logic controllers do
    is that fuzzy control rules are just providing a simple and efficient
    interpolation method. A set of fuzzy rules is just describing a set of
    (fuzzy) points (x,y) in a multi-dimensional space belonging to an
    "unknown" graph. Then, given an input, the output is computed by applying
    a simple analogical reasoning principle: the closer the input is to x, the
    closer the output will be to y. That's all. But whether this has to do
    with logic or not it is matter of current discussions.

    Lluis.

    --------------------------------------
    Lluis Godo
    IIIA - CSIC
    Campus Univ. Autonoma de Barcelona s/n
    08193 Bellaterra, Spain
    --------------------------------------



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