Re: Imprecise Probability

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Sat Dec 16 2000 - 09:31:23 MET

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    Otto Cordero wrote:
    >
    > Hello
    >
    > I would like to be more specific about my previous question....
    > Lets supose someone is asked to give his estimation about the probability of
    > some event, answers like "0.78 of probability" are very unlikely. I would
    > expect something as "high probability" or "low probability", here we
    > linguistic terms, wich could be associated with fuzzy sets.
    > I am asking for more information on how fuzzy sets theory joins with
    > probabilty theory to manage this kind of situations.

    If I may be so bold, I would suggest you take a look at my _Fuzziness
    and Probability_ (1995), available from Amazon.Com. Here is an
    unsolicited testimonial ca. 1996 from someone I've never met:
    +----------------------------------------------
    |JG.Campbell (jg.campbell@ulst.ac.uk) wrote:
    |
    |<snip>
    |
    |: I've grappled with "fuzziness vs probability" for quite a few years
    |: and, in my opinion, S.F. Thomas' book is the closest yet to a
    |: self-contained and reasoned, logical discourse on the matter. Yes, it
    |<snip>
    |
    |What a lovely ego boost! I do appreciate the very kind remarks...
    |
    |: Regards,
    |: Jon Campbell
    |
    |Best,
    |S. F. Thomas
    +---------------------------------------------
    The original book announcement for Fuzziness and Probability can still
    be found on the web at
    http://www.science.at/marchives/fuzzy-mail95/1264.html .

    I remember asking much the same question as Otto is now asking back in
    1977 when I commenced my Ph.D. program. I was taking a course in
    Bayesian probability and statistics at the same time that I was taking a
    course in fuzzy set theory. The question that occurred to me was, if we
    tossed a thumb tack, and we judged that there was a "high" prior
    probability that the tack would fall head down rather than on its side,
    how would one use a Bayes-like procedure to reduce the (fuzzy)
    uncertainty in that initial probability estimate, based on repeated
    actual observations of the outcome on repeated tossings. The result was
    my 1979 University of Toronto dissertation entitled "A Theory of
    Semantics and Possible Inference, with Application to Decision
    Analysis". Many years later, in 1995 as indicated, I finally got it
    published in book form. As my career took me out of the Academy, I have
    been unable to follow developments as closely as I would like. Still, it
    does not escape me that every new generation of students of fuzzy theory
    continue to ask the same question that occurred to me so very long ago,
    suggesting that the mainstream theory has not yet fully or adequately
    addressed the question. There are good "sociological" reasons why that
    is so, as the early proponents of fuzzy have sought mightily to distance
    the fuzzy theory from the probability theory, fearing that if that were
    to happen, it would be forever discounted as "nothing new". We are past
    that stage now, I believe. Fuzzy is now a successful, mature discipline.
    We can go back and clean up the fundamentals, and in the process, the
    answer to the kind of question that Otto has raised would become crystal
    clear. At any rate, that is what _Fuzziness and Probability_ sets out to
    do. In the process, it also cleaned up some errors in the fundamentals
    of Bayesian inference, although that was not what I set out to do. At
    any rate, I believe it succeeds, and unsolicited testimonials of the
    sort quoted above strengthens me in that belief. You may want to check
    it out. You may also want to take a look at various discussions in the
    fuzzy archives where the sort of issue you have raised has been
    discussed again and again. In particular, take a look at
    http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail96/ , also the archives
    for 95 and 97, and do a search on "probability". You will see some
    interesting discussion on the intersection of fuzzy and probability,
    including from my good net friends Ellen Hisdal and W. Siler as well as
    myself.

    > Thanks a lot.
    >
    > Otto

    Regards,
    S. F. Thomas

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