Dear Lotfi, Thanks very much for your prompt and thoughtful response.
> Concretization in terms of membership functions is simplest and >most natural. Some people find membership functions simple and natural. Others don't. >If u is a real number, than the grade of membership of u in the >fuzzy set "approximately a" is simply the subjective degree to which >u fits your intended meaning of "approximately a." I don't know what that means. I can operationalize a subjective degree of belief in a number of different ways. In Morris DeGroot's axiomatization, for example, he assumes the existence of a uniform randomly distributed variable (e.g., an idealized spinner that has equal probability of landing at any point around the circumference of a perfect circle). I compare an event I am thinking about to the spinner and imagine a pie-shaped wedge colored gray. I adjust the gray-colored area until I judge that the likelihood of the spinner landing in the gray area is the same as the likelihood of the uncertain event. There are two observable, measurable events: the event whose likelihood I am judging and the spinner result, and my task is to compare their likelihoods. There is a clear meaning to this task. Thus, there is a clear operational definition of precisely what your thought experiment entails. We assign a prior distribution for p, reflecting the decision maker's beliefs about the probability given nothing other than the context in which elicitation is occurring. Then we assign a likelihood to obtaining the evidence "p is approximately a" given each value of p, again in the context in which elicitation is occurring. Presumably, this likelihood would be near zero for values of p far from a, and would peak somewhere around a. We condition on this evidence and update our beliefs for p. Apparently you find fuzzy membership more intuitive than this. I don't. I simply have no idea how to construct an operational definition of the subjective degree to which p fits my intended meaning of "approximately a." >If you interpret "approximately a" as a probability distribution, >then various problems arise, among which is the problem of the >meaning of conjunction. Please explain what you mean by the problem of the meaning of conjunction. > The crux of the difficulty in my example is that there are three >distinct goals: (a) maximize entropy; (b) satisfy the fuzzy >constraint on the mean; and (c) satisfy the fuzzy constraint on the >variance. Thus, what you have is a problem in multicriterion >optimization. There is no consensus on how such problems should be >solved. The decision theoretic consensus on multicriterion optimization is that the agent doing the optimizing needs to specify a utility function that reflects his or her priorities among the different criteria. In decision theory, there is consensus on the approach, but one would not expect consensus on the result, because naturally different agents would be expected to have different priorities. My suggested semantics would also not necessarily lead to consensus on the answer. Different agents would have different prior beliefs and different likelihood functions. We would therefore expect a range of posterior distributions. This is OK in subjectivist theory. If we had a great deal of statistical data, the subjective distributions of all but the truly dogmatic agents would reach consensus that the value is very close to the observed relative frequency. But this is as it should be. Given an under-constrained problem that has no "correct answer," one would not expect consensus on an answer. >That is why I raise the question of how can the maximum entropy >principle be applied when the side-conditions are imprecisely >defined. You have my suggestion. Kathy
