From: owner-uai@cs.orst.edu on behalf of Lotfi A. Zadeh
[zadeh@cs.berkeley.edu]
Sent: 17 lipca 2003 02:34
To: uai@cs.orst.edu
Subject: Re: [UAI] Maximum Entropy Principle

Dear Kathy:

	In your thought-provoking comment you touch upon a number of basic 
issues. In my response, I will focus on just a few.

	Underlying our exchanges is a basic difference in our views regarding the sufficiency of standard probability theory, call it PT, to deal with uncertainty and imprecision. In your view, which at present is an overwhelmingly majority view, PT is sufficient. In my view, which at present is an overwhelmingly minority view, it is not. An exposition of my view may be found in the paper entitled "Toward a Perception-Based Theory of Probabilistic Reasoning with Imprecise Probabilities," which appeared in a special issue on imprecise probabilities of the Journal of Statistical Planning and Inference, Vol. 105, pp. 233-264, 2002, (downloadable at http://www-bisc.cs.berkeley.edu/BISCProgram/Projects.htm). In the preface to the issue, the co-editor, Professor Jean-Marc Bernard, has this to say:

	"There is a wide range of views concerning the sources and significance of imprecision. This ranges from de Finetti's view, that imprecision arises merely from incomplete elicitation of subjective probabilities, to Zadeh's view, that most of the information relevant to probabilistic analysis is intrinsically imprecise, and that there is imprecision and fuzziness not only in probabilities, but also in events, relations and properties such as independence. The research program outlined by Zadeh is a more radical departure from standard probability theory than the other approaches in this volume."

	Please note that my critique of PT is constructive in the sense that what is suggested in my JSPI paper is a generalization of PT which can and should enhance the ability of probability theory to deal with real-world problems.

	The point of departure on my approach is the observation that there are two concepts which play a key role in human cognition, partiality and granularity. Partiality relates to the fact that most human concepts are partial in the sense that they are associated with a scale, that is, are a matter of degree. Thus, we have partial understanding, partial knowledge, partial similarity, partial truth, partial certainty and partial possibility, with the last three standing out in importance.  Furthermore, reflecting the bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information, the scale is granulated, with a granule being a clump of values drawn together by indistinguishability, similarity, proximity and functionality. For example, the granules of likelihood might be likely, unlikely, very unlikely, etc.

	Standard probability theory, PT, addresses partiality of certainty, but what it does not address is partiality of truth and partiality of possibility. As a consequence, PT does not have the capability to deal with perceptions, which are intrinsically imprecise and, in general, involve partiality of certainty, truth and possibility. For example, the perception: It is very unlikely that it will be a warm day tomorrow, involves an imprecise perception of likelihood and an imprecise perception of temperature.

	Standard probability theory provides no machinery for (a) representing the meaning of perceptions described in a natural language; and (b) reasoning with them. An example is: Usually Robert leaves office at about 6 pm and arrives at home about half an hour later. When does Robert arrive at home?

	Perceptions, and especially perceptions of likelihood, are intrinsically imprecise. This reality is overlooked in subjective probability theory--a theory in which subjective probabilities are assumed to be crisply defined. Thus, if I am asked, "What is the probability that tomorrow will be a warm day," then using the analogy of the spinner, my perception would correspond to a wedge with fuzzy rather than crisp edges. In other words, my perception would be a fuzzy rather than crisp probability. (See the book "Fuzzy Probabilities," by J.  Buckley, Springer-Verlag, 2003.) What this means is that axiomatization of subjective degrees of belief cannot be accomplished within the conceptual structure of bivalent logic.

	With regard to the maximum entropy principle, you seem to agree that maximization subject to imprecise constraints involves a great deal of arbitrariness. As a consequence, the uniqueness of entropy-maximizing distribution is lost.

	In summary, what is almost universally unrecognized is that standard probability theory has fundamental limitations which are rooted in the use of bivalent logic as its foundation. Abandonment of bivalence is a prerequisite to enhancing the power of probability theory to deal with real-world problems.

		With my warm regards,

			Lotfi
--
Professor in the Graduate School, Computer Science Division Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720 -1776 Director, Berkeley Initiative in Soft Computing (BISC)