Herman Rubin wrote:> > The fuzzicists CLAIM to have an alternative to probability, > but it a mistake in logic which has often been repeated.
The ones that do so are misguided in my opinion. I for one explicitly claim no such thing, rather a duality relation between fuzziness and probability exactly akin to that between probability and likelihood, and for similar reasons. Probability (over sample space) induces likelihood (over parameter space). The direction of "induction" in this sense goes the other way as well, but I leave that aside. It is the same with fuzziness, or what could be called semantic likelihood. The fact that language-use is a chance phenomenon (would a randomly selected competent speaker use the word "tall" to describe a given height value, yes or no, is a Bernoulli process) induces a fuzzy range of possible values for what any given speaker could mean when he uses the word tall. Fuzziness and probability are thus inextricably linked in my opinion, and neither can be an alternative to the other, any more than that likelihood could be an alternative to probability. They are dualistic flip sides of the same uncertainty coin.> The implication many make is that probability is a truth > value in the sense of logic; it is not. A truth value > system has the property that the truth value of a compound > proposition is determined from that of the components. > The truth value system used in probability is that of a > Boolean algebra, and probability is a functional on it > which has certain properties, which happen to make if of > some utility in application. > > Now what happens with fuzzy logic? If the fuzzicists > were to assign values to compound propositions in a > consistent way, they would end up with probability. > If the do not, how can someone decide on a course of > action under uncertainty?
Respectfully, I think you are wrong here. But I cannot attempt a showing unless you demonstrate what you mean by "consistency" in this context. There may be an additivity issue that applies for probability but not for fuzzy, or likelihood, which might explain your claim to the "inconsistency" of the latter.> BTW, the behavioristic Bayesian approach does not > start out with the prior as probability, but merely > as the measure corresponding to a positive linear > operator on utility functions, and it is only the > product of loss and prior which has implications for > the course of action.
I really don't understand this. The allusion here is to a development that must have taken place sometime after I bailed out of the Bayesian boat. But if I undertand it correctly, it seems to imply that the data are irrelevant, which would be a puzzling position for even a committed Bayesian to take.> -- > This address is for information only. I do not claim that these views > are those of the Statistics Department or of Purdue University. > Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 > hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Regards, S. F. Thomas