Re: Inevitable Illusions

From: Sidney Thomas (sf.thomas@verizon.net)
Date: Sun Jun 10 2001 - 13:33:46 MET DST

  • Next message: Nico du Bois: "Re: fuzzy proofs and law of excluded middle"

    > From: Chris De Voir (devoirc@biotronik.com) Message 1 in thread
    > Subject: Inevitable Illusions
    > Newsgroups: comp.ai.fuzzy
    > Date: 2001-06-01 02:28:24 PST
    >
    > Thank you for your comments and clarifications. But there is a particular
    > slant to my line of questioning that I am trying to convey. I offer another
    > example found at:
    > http://www.dcs.qmw.ac.uk/~norman/BBNs/Representativeness.htm (where there
    > are more examples relevant to thread of my original question).
    >
    > "Insensitivity to prior probability of outcomes
    >
    > Suppose you are given the following description of a person:
    >
    > 'He is an extremely athletic looking young man who drives a fast car and has
    > an attractive blond girlfriend.'
    >
    > Now answer the following question:
    >
    > Is the person most likely to be a premiership professional footballer or a
    > nurse?
    >
    > If you answered professional footballer then you were sucked into this
    > particular fallacy. You made the mistake of ignoring the base-rate
    > frequencies of the different professions simply because the description of
    > the person better matched the stereotypical image. In fact there are only
    > 400 premiership professional footballers in the UK compared with many
    > thousands of male nurses, so in the absence of any other information it is
    > far more likely that the person is a nurse."
    >
    > In light of this example, the questions I am really trying ask are:

    I take the general point about the application of Bayes theorem,
    which it seems to me is unobjectionable in _this_ context. (There is
    another context, well-loved by Bayesians, which in my opinion cannot
    be justified, namely the treatment of model parameters as though they
    were random variables, and the invocation of a subjectivist alchemy
    to achieve this unwarranted transformation. In the present example,
    this issue does not arise, and the application of Bayes theorem is
    wholly acceptable.) However, the point of principle sought to be made
    is clouded by some of the specifics in the example, and the
    conclusion sought is far from certain. Premiership football players
    in the UK are certain to be "extremely athletic looking and young",
    while the same cannot be said for male nurses. It is therefore not
    enough to know that male nurses outnumber premiership football
    players by a wide margin in order to reach the conclusion sought (and
    to make the corresponding point); one must also know the fraction of
    male nurses a) who are young, _and_ b) "extremely athletic looking",
    _and_ c) have income sufficient to afford a "fast car", _and_ d) are
    attractive enough themselves to win the favor of an attractive
    girl-friend. The proper application of Bayes theorem, in the present
    example, would seem to depend on rather more than just relative
    counts of male nurses and premiership footballers.

    > 1. Would not Fuzzy thinking yield an answer that would be consistent with
    > (in the same ballpark as) what Kahneman & Tversky say is the correct answer?

    The issues raised in this problem are not fuzzy, except in a
    second-order sense. Bayes theorem may properly be applied, but to the
    extent some of the relevant probabilistic events are fuzzy --
    "extremely athletic", "young", "attractive", etc. -- it is clear (I
    think) how the fuzzy set theory may in principle be applied, since it
    allows the evaluation of the probability of a fuzzy event. In this
    way however, it serves to extend the probability calculus rather than
    to compete with it.

    > 2. What "systeme" in Fuzzy assures this? In another post to this list,
    > "Bayes-learning-thought etc", Martin Lefley (Thu Jan 18 2001) stated,
    > "Bayesian reasoning is represented by formulae that could be represented
    > by...FLS...." Does anyone have a pointer to this method?
    >
    > 3. Could Fuzzy thinkers come up with answers that Kahneman & Tversky would
    > identify as heuristically biased? This is not a rhetorical question, since
    > #4 and #5 follow.
    >
    > 4. In that case, what essential(s) of the Fuzzy reasoning process has been
    > overlooked or misapplied?
    >
    > 5. In the case of #3 as an outcome, could it be that no violations Fuzzy
    > reasoning have occurred?

    You raise a _lot_ of questions here, perhaps more than you know, and
    I'm not sure to what, specifically, you allude. However, I would
    hazard that what you may be after is that there is an area where
    fuzzy and Bayesian are in competition, and indeed conflict, to which
    _I_ have alluded above. It is the same area where Bayesian inference
    and classical statistical inference are in disagreement, namely over
    the logical status of model parameters. Classical statistical
    inference is clear that they are not random variables, and rests upon
    a considered rejection of an application of Bayes theorem that treats
    uncertainty in model parameters symmetrically with uncertainty of
    outcome in an observable chance phenomenon. The axioms and theorems
    of subjective probability, heroic as they are, are insufficient to
    make a true random variate out of a model parameter, and in my
    opinion fail even as metaphor. The Bayesian inferential setup asserts

            b(w|x) ~ L(w;x) * b(w)

    where b(w|x) is the posterior distribution of the parameter w given
    the observation x, L(w;x) is the likelihood function of the parameter
    w consistent with the observation x, b(w) is a subjectively injected
    prior distribution of the parameter w, and the symbol ~ represents
    similarity -- multiplication (or division) by an appropriate constant
    is necessary to yield the correct distribution that must integrate to
    unity, otherwise we are simply multiplying the prior by the
    likelihood to get the posterior. Clearly, if we are concerned to
    characterize only what the *data* say, disregarding any subjective
    prior (or prejudice) to which one might admit, the focus must be on
    the likelihood function. But as every schoolboy knows, and as Fisher
    made clear when he developed the concept, the likelihood function is
    not a probability distribution, and it certainly requires a
    subjective alchemy to turn it into one. At any rate, I have argued in
    my _Fuzziness and Probability_ (ACG Press, 1995) that likelihood
    represents uncertainty of a possibilistic rather than probabilistic
    sort, and that prior and posterior uncertainty about a model
    parameter must also, to be correct, be represented possibilistically.
    Once it is appreciated that the likelihood function of statistical
    inference falls within the ambit of a possibility theory, albeit one
    that needs to differ from the Zadehian in one small but significant
    way, it is but a short step to the extended likelihood calculus that
    eluded Fisher and the generations of statisticians since. And once we
    have an extended likelihood calculus, the only remaining
    justification for the improper Bayesian stratagem, of treating
    uncertainty about model parameters symmetrically with uncertainty
    about the random variables whose distributions they model...namely
    the possibility it opens up for a direct characterization of the
    former...is rendered moot, since the extended likelihood (or
    possibility) calculus allows the same benefit, and without the
    stretch of metaphor, albeit with very sophisticated mathematics, in
    which subjective Bayesianism must indulge.

    Perhaps it is something along this line that you were striving to
    elicit.

    > Chris.
    >

    Regards,
    S. F. Thomas

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