RUSPINI@IU.AI.SRI.COM (Enrique Ruspini) (03/23/88)
It is usually very difficult (as well as annoying) to engage on another round of the "acrimonious" debate on approximate reasoning since often the discussion deals with side-issues or supposed "paradoxes" or "inconsistencies" of theories that require, for proper understanding, a good deal of sophistication. The contribution of Star to AILIST (10 Mar 88) provides, however, reasonable grounds for discussion as Star makes reasonably succint points describing the bases for his subjectivist preferences. Unfortunately, that conciseness is not matched by solid scientific arguments. Before analyzing his four purportedly unique characteristics of the subjectivist approach, let me just say that it is plain wrong to consider fuzzy sets as an alternative to either Dempster-Shafer or classical probability. It is actually an approach that complements probabilistic reasoning by providing another type of insight on the state of real world systems. If, for example, we say that the probability of `"economic recession" is 80%' we are indicating that there is either a known (if we are thinking of probabilities in an objective sense) or believed (if take a subjectivist interpretation) tendency or propensity of an economical system to evolve into a state called "recession". If, on the other hand, we say that the system will move into a state that has a possibility of 80% of being a recession, we are saying that we are *certain* that the system will evolve into a state that resembles or is similar at least to a degree of 0.8 (in a preagreed scale) to a state of recession (note the stress on certainty with imprecision about the nature of the state as opposed to a description of a believed or previously observed tendency). Clearly, possibilistic and probabilistic approaches have different epistemological bases. To try to force a unique view on uncertain reasoning (and on all interpretations of probability) as suggested by some is a bit similar to trying to understand the physics of light using solely wave-based models. Passing now to the unique distinguishing characteristics of "mainstream" approaches let us take them one by one: > Subjective probability ...is the only approach that has the > characteristics of > 1. Being based on a few, acceptable simple axioms. The multiple assertions contained in this first statement are all either wrong or, at best, rather subjective matters. Wheter or not Savage's axioms (Foundations of Statistics) are acceptable (whatever that means) or sufficient is arguable. Readers may find it interesting to look at Savage's axioms and decide by themselves if they are either "simple" or "acceptable". The insufficiency of such systems to capture enough aspects of rational behavior, for example, has been criticized (e.g.,Kyburg,H., J. Phil., 1974). It is interesting to note also that some of these systems (e.g., Cox) contain axioms that many find objectionable because they appear to have been introduced solely to validate certain characteristics of the approach (e.g., Bayes conditionalization). So much for acceptability. As for simplicity it is interesting to note that if one does away with some of the axioms in systems such as Savage or Cox one immediately gets systems (which are, necessarily, simpler) characterized by interval- (rather than number-) valued probabilities. See, for example, the critique of Savage axioms in "Suppes, P., The Measurement of Belief, J. Roy. Stat. Soc., 1974. There are also enough paradoxes around showing that rather rational folk (including prominent subjectivists) often engage in behavior which is inconsistent with their own prescriptions (e.g., the well-known "Ellsberg" and "Allais" paradoxes). Of course one can define "rational behavior" any way one wants and declare that even oneself is not always rational but the shortcomings of these tricks in word-play are clear: one should prescribe standards for rationality and find out if one's recipe always assures compliance with them, rather than define "rational behavior" solely as that which comes out of following one's preferred procedures (which are far from being noncontroversial) !!! To end this point it is important to note that axiomatizations of fuzzy sets and D/S exist (including recent development of model-theoretic semantics for them --- more below on this) although I must admit that I feel that it is rather silly to try to defend or attack approaches on such bases: often systems (including those used to reason) are very complex and it is ridiculous to expect that formal systems will be capable of capturing such complexity using straightforward formalisms. Under such conditions, simplicity should be a reason for suspicion and concern. > 2. Being able to connect directly with decision theory > (Dempster-Shafer can't). This is nonsense. Interval probability approaches (of which D/S is an example) may be applied to decision analysis by simply extending the techniques proposed by subjectivists. The difference (and I suspect that this is what Star means by "connection") is that by producing intervals of expected values for each decision they fail to tell sometimes if a decision is better or worse than another. There is nothing wrong with this, though: all that the results tell you is that there are neither rational bases nor factual data to assure you that A is preferrable to B, a true fact of life. Insisting that one approach is better because it always produce a needed decision (the famous "pragmatic necessity") even when the factual support is not there leaves one wondering about such a choice of standards (if something must be done, then an interval-based approach followed by coin flipping will produce results that are as "reliable" or "rational"). It is interesting to note that if the word "belief" is replaced by "temperature" in some of these curious epistemological arguments, then we may convince ourselves that we always know the temperature of everything (or act as if we do) and decide to do away with thermometers. Readers may also wonder about the curious circularity of support in the pairs of arguments: "We always know degrees of belief about any proposition (or act as if we do) because in the end we always do something" and "We do something that is best because we always have rationally derived degrees of belief." It is also interesting to wonder whether Bayesian decision theory (based primarily on the notion of expected utility as the unique performance measure for decisions) is sufficient for the variety of complex problems found on modern applied science (e.g., "What is the sense of choosing business strategies that are only good in the long-run when a (single) unfortunate turn of events may leave us broke?"). > 3. Having efficient algorithms for computation. I do not know where Star got this notion. Efficient algorithms are available for both D/S and fuzzy sets. Being present at the last AAAI in Seattle, I recall that Judea Pearl mentioned the great computational and formal similarities between some of the Bayesian and D/S network algorithms (e.g., some of Pearl's algorithms and the procedures for evidence combination of Dempster/Kong). It is difficult to believe that if a prominent Bayesian makes such an assessment, one class of procedures could be efficient while the other is not (of course, efficiency per se is of little value if your method gives you the wrong solution!) As for fuzzy sets, their very purpose is to simplify the analysis of complex systems by trading unneeded precision for increased depth of understanding. AIListers may be interested to know more about an operational subway system control in Japan (in the city of Sendai) designed by Hitachi, over a reported 8 years period, which uses fuzzy logic. Works describing this effort (Proceedings IFSA 1987) indicate also the reasons why fuzzy logic was used over other alternatives (anybody with a background in control theory will shudder at the complexities of handling --even if they were known-- the required probabilities, covariance matrices, etc. involved in a classical stochastic control approach for large complex, control systems!). 4.> Being well understood. I do not know whether Star is criticizing other theories (which had been studied only for a few years) for not being as well understood as classical probability . Setting aside whether or not probability (particularly in its subjective interpretation) is "well understood" (a matter much disputed among philosophers of probability), I do not feel that it is particularly surprising that recent technological developments are not as well developed or understood as techniques that have been around for a long while (One can only wonder why this make the old techniques more acceptable to deal with new classes of problems). If one looks at the applicability and current rate of progress in new approaches, however, one sees a different story. Both fuzzy sets and D/S are advancing strongly at the theoretical and applied level. Dempster/Shafer has been found to have solid foundations rooted in classical probability and epistemic logic (Ruspini, Proc. 1987 IJCAI; SRI Technical Note 408). Recently formal semantics have been developed for both fuzzy sets and D/S. [Ruspini, IPMU 1988]. Readers may want to contrast this vigorous advance with subjectivists, who, after 40 or 50 years, have failed to generate convincing arguments supporting their conceptual reliance on algorithms that assume that we always know probabilities of events (or act as if we do) even when rational or empirical bases to support such knowledge are admittedly absent ! I do not know where Star is looking ("Look at what people are doing with Dempster-Shafer belief functions or fuzzy sets.") but I, for one, have looked enough, and with a considerable background in formal sciences, I do not see anything that brings to mind the images that these approaches evoke in Star's mind ("mainstream approaches" versus "more experimental"). To repeat myself, it is ridiculous to expect to find strong formalisms around newly evolving theories (Would anybody expect Galilean mechanics to have been formalized before being seriously considered?). The state of development of both D/S and fuzzy sets is neither unfounded nor solely "experimental" (a questionable epithet) or non "mainstream" (another convenient but inaccurate qualifier), however. One should be very leery, on the other hand, of questionable practices that purport to derive "rational" decisions in the absence of knowledge: a miracle that I have described elsewhere (Comp. Intelligence, February 1988) as epistemological alchemy. I would like to say that my comments should not be construed to negate the value of classical probability in the study of uncertainty. Furthermore, I believe that it is important to continue to study the concept of belief and the problems associated with its quantification and recognize the positive contributions that subjectivists have made (and, undoubtely, will continue to make) to the art and science of probabilistic reasoning. I believe, however, that, while striving to improve existing methodologies, however, we should keep an open mind towards novel concepts while realizing that the former approaches might not be (at least yet!) as comprehensive and efficient as some zealously purport them to be. -------