Robert Ehrlich <bobehrlich@home.com> writes:> IMHO fuzzy memberships reflect the degree of hybridness of samples and > have nothing especially to do with prob.
But the only concrete suggestion you see in fuzzy logic books for how to obtain fuzzy memberships numbers is to use the proportion of domain experts who say the man is tall, or whatever. Thinking through the implications (of issues like: variability due to the expert set being finite), my intuition is led quickly to consider it a likelihood, making each expert's opinion very like an uncertain measurement, if you are thinking in Bayesian terms. In other words, exactly Thomas's "semantic likelihood". Except, I don't understand his problem with "the vain Bayesian attempt to treat likelihood as though it were probability". Bayes treats likelihoods as conditional pdfs, and that is a powerful way of looking at problems involving measurements, observations and so on. One of the big questions which FL attempts to answer is "what is it that I can conclude from hearing someone make a vague assertion like `X is tall', and how can we represent it scientifically?" Bayes has an answer to that already, in terms of [my beliefs about] the speaker's utterance disposition: i.e. what I expect them to say under certain circumstances, i.e. the conditional probability that they will assert tallness of X given [their beliefs about] X's numerical height. This does the job: it answers the question, and in terms uniform with those offered by Bayes for other "uncertainty" issues. In short it seems to me that Thomas over-stresses the value added by FL. That FL is (afaics) a "mere" branch of Bayesian probability is not _so_ strange if you recall how committed Bayesians consider probability as akin to logic, and in particular how they often think of log probabilities as "degrees of surprisal". A formula like p(tall | height=1.92) = 0.2 reads better if you think "how surprised would one be to hear a 1.92-high person decribed as tall?". Disclaimer: I am not any kind of expert, I just haven't ever found an FL devotee who was able seriously to engage with these points, so I'd love to hear more from Thomas.> There is plenty new semantics in the fuzzy set theory > of which probabilists have been blissfully unaware, and which in > fact helps to illuminate some problems in the foundations at > least of statistical inference theory.
Specifically?