Well, I can't even be sure that the set A includes Robert, since Robert may not even use a multiplication table to decide on each question. (He could be autistic and very quickly add the numbers together each time, never memorizing the table). Bayesians care a lot less about membership in A than in the probability that Robert will get the next question right, given he answered the test with X% accuracy.
Bob Welch -----Original Message----- From: Andrzej Pownuk [mailto:[EMAIL PROTECTED] Sent: Monday, July 28, 2003 1:40 PM To: [EMAIL PROTECTED] Subject: RE: [UAI]Maximum Entropy Principle I have only small comment to the message which is given below. Let us consider the following problem. There is a set of pupils X={John, George, Steven} and there is a test which checks the knowledge of multiplication table. Let us consider the following results of the test: John - 0% (correct answers) George - 30% (correct answers) Steven - 60% (correct answers) Robert - 100% (correct answers) Now let's try to create the following sets: A - a set of persons who knows multiplication table. B - a set of persons who doesn't know multiplication table. As we can see A={Robert}, B={John}. Well but what about George and Steven? Does George and Steven know the multiplication table or not? It is difficult to answer to this question using "Yes" or "NO" answers. Because of that at school teachers do not use only "Yes" and "No" statements. Teachers use "degree of knowledge" (i.e. degree). This is quite natural solution and everybody knows this problem. The sets A and B have no crisp boundary or in other words it is more convenient to use degree instead of "Yes" and "No" statements. Except the situation when people are drunk the knowledge of multiplication table is non-random (in short period of time). Then I have no idea how to define the sets A and B using probability theory (and because of that also Bayesian methods). I don't know how to call those phenomena. Maybe this is fuzzy set maybe not. However, if we follow this way of thinking then it is very easy to prove that in general t-norm cannot be apply to this kind of "grade of membership". Because of such problems I don't know why t-norm can be applied to other grades of membership (i.e. to completely subjective answer to the question "How well x belong to the set A?".) The final conclusion: Non-crisp sets are used in everyday life very often and they are completely no-random but I don't think that current fuzzy logic describes the problem well. Regards, Andrzej Pownuk - --------------------------------- Ph.D., research associate at: Chair of Theoretical Mechanics Faculty of Civil Engineering Silesian University of Technology URL: http://zeus.polsl.gliwice.pl/~pownuk - - ---------------------------------
