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*> From: Chris De Voir (devoirc@biotronik.com) Message 1 in thread
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*> Subject: Inevitable Illusions
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*> Newsgroups: comp.ai.fuzzy
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*> Date: 2001-06-01 02:28:24 PST
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*>
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*> Thank you for your comments and clarifications. But there is a particular
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*> slant to my line of questioning that I am trying to convey. I offer another
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*> example found at:
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*> http://www.dcs.qmw.ac.uk/~norman/BBNs/Representativeness.htm (where there
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*> are more examples relevant to thread of my original question).
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*>
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*> "Insensitivity to prior probability of outcomes
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*>
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*> Suppose you are given the following description of a person:
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*>
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*> 'He is an extremely athletic looking young man who drives a fast car and has
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*> an attractive blond girlfriend.'
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*>
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*> Now answer the following question:
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*>
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*> Is the person most likely to be a premiership professional footballer or a
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*> nurse?
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*>
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*> If you answered professional footballer then you were sucked into this
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*> particular fallacy. You made the mistake of ignoring the base-rate
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*> frequencies of the different professions simply because the description of
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*> the person better matched the stereotypical image. In fact there are only
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*> 400 premiership professional footballers in the UK compared with many
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*> thousands of male nurses, so in the absence of any other information it is
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*> far more likely that the person is a nurse."
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*>
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*> In light of this example, the questions I am really trying ask are:
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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
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*> (in the same ballpark as) what Kahneman & Tversky say is the correct answer?
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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,
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*> "Bayes-learning-thought etc", Martin Lefley (Thu Jan 18 2001) stated,
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*> "Bayesian reasoning is represented by formulae that could be represented
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*> by...FLS...." Does anyone have a pointer to this method?
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*>
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*> 3. Could Fuzzy thinkers come up with answers that Kahneman & Tversky would
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*> identify as heuristically biased? This is not a rhetorical question, since
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*> #4 and #5 follow.
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*>
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*> 4. In that case, what essential(s) of the Fuzzy reasoning process has been
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*> overlooked or misapplied?
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*>
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*> 5. In the case of #3 as an outcome, could it be that no violations Fuzzy
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*> reasoning have occurred?
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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.
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*>
*

Regards,

S. F. Thomas

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