STAR@LAVALVM1.BITNET (Spencer Star) (03/31/88)
I'll try to respond to Ruspini's comments about my reasons for choosing the Bayesian approach to representing uncertainty. [AIList March 22, 1988] >If, for example, we say that the probability of "economic >recession" is 80%, we are indicating that there is either a >known...or believed...tendency or propensity of an economic >system to evolve into a state called "recession". > >If, on the other hand, we say 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 >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). I think this is supposed to show the difference between something that probabilities can deal with and a "fuzzy approach" that probabilities can't deal with. However, probabilities can deal with both situations, even at the same time. The following example demonstrates how this is done. Suppose that there is a state Z such that from that state there is a P(Z)=60% chance of a recession in one year. This corresponds to the state in the second paragraph. Suppose that there are two other states, X and Y, that have probabilities of entering a recession within a year of P(X)=20% and P(Y)=40%. My beliefs that the system will enter those states are Bel(X)=.25 Bel(Y)=.30 Bel(Z)=.45, where beliefs are my subjective prior probabilities conditional on all the information available to me. Then the probability of a recession is, according to my beliefs, P(recession)=alpha*[Bel(X)*P(X) + Bel(Y)*P(Y) + Bel(Z)*P(Z)], where alpha is a normalization factor making P(recession) + P(no recession) = 1. In this example, a 25% belief in state X occurring gives a 5% chance of recession and a 20% chance of no recession. Summing the little buggers up across states gives a 44% chance of recession and a 56% chance of no recession with alpha=1. These latter figures give my beliefs about there being or not being a recession. So far, I haven't found a need to use fuzzy logic to represent possibilities. Peter Cheeseman, "Probabilistic versus Fuzzy Reasoning", in Kanal and Lemmer, Uncertainty in AI, deals with this in more detail and comes to the same conclusion. Bayesian probabilities work fine for the examples I have been given. But that doesn't mean that you shouldn't use fuzzy logic. If you feel more comfortable with it, fine. I proposed simplicity as a property that I value in choosing a representation. Ruspini seems to believe that a relatively simple representation of a complex system is "reason for suspicion and concern." Perhaps. It's a question of taste and judgment. However, my statement about the Bayesian approach being based on a few simple axioms is more objective. Probability theory is built on Kolmogoroff's axioms, which for the uninitiated say things like the sum of the probabilities must equal 1, that we can sum probabilities for independent events, and that probabilities are numbers less than or equal to one. Nothing very complicated there. Decision theory adds utilities to probabilities. For a utility function to exist, the agent must be able to order his preferences, prefer more of a beneficial good rather than less, and a few other simple things. Ruspini mentions that "rational" people often engage in inconsistent behavior when viewed from a Bayesian framework. Here we are in absolute agreement. I use the Bayesian framework for a normative approach to decision making. Of course, this assumes the goal is to make good decisions. If your goal is to model human decision making, you might very well do better than to use the Bayesian model. Most of the research I am aware of that shows inconsistencies in human reasoning has made comparisons with the Bayesian model. I don't know if fuzzy sets or D-S provides solutions for paradoxes in probability judgments. Perhaps someone could educate me on this. >Being able to connect directly with decision theory. >(Dempster-Shafer can't) Here, I think I was too hard on D-S. The people at SRI, including Ruspini, have done some very interesting work using the Dempster-Shafer approach in a variety of domains that require decisions to be made. Ruspini is correct in pointing out that if no information is available, the Bayesian approach is often to use a maximum entropy estimate (everything is equally likely), which could also be used as the basis for a decision in D-S. I have been told by people in whom I trust that there are times when D-S provides a better or more natural representation of the situation than a strict Bayesian approach. This is an area ripe for cooperative research. We need to know much more about the comparative advantages of each approach on practical problems. >Having efficient algorithms for computation. When I stated that the Bayesian approach has efficient algorithms for computation, I did not mean to state that the others did not. Shafer and Logan published an efficient algorithm for Dempster's rule in the case of hierarchical evidence. Fuzzy sets are often implemented efficiently. This statement was much more a defence of probability theory against the claim that there are too many probabilities to calculate. Kim and Pearl have provided us with an elegant algorithm that can work on a parallel machine. Cycles in reasoning still present problems, although several people have proposed solutions. I don't know how non-Bayesian logics deal with this problem. I'd be happy to be informed. >Being well understood. This statement is based on my observations of discussions involving uncertainty. I have seen D-S advocates disagree numerous times over whether or not the particular paper X, or implementation Y is really doing Dempster-Shafer evidential reasoning. I have seen very bright people present a result on D-S only to have other bright people say that result occurs only because they don't understand D-S at all. I asked Glen Shafer about this and his answer was that the theory is still being developed and is in a much more formative stage than Bayesian theory. I find much less of this type of argument occurring among Bayesians. However, there is also I.J. Good's article detailing the 46,656 varieties of Bayesians possible, given the major different views on 11 fundamental questions. The Bottom Line There has been too much effort put into trying to show that one approach is better or more general than the other and not enough into some other important issues. This message is already too long, so let me close with what I see as the major issue for the community of experts on uncertainty to tackle. The real battle is to get uncertainty introduced into basic AI research. Uncertainty researchers are spending too much of their time working with expert systems, which is already a relatively mature technology. There are many subject areas such as machine learning, non-monotonic reasoning, truth maintenance, planning, etc. where uncertainty has been neglected or rejected. The world is inherently uncertain, and AI must introduce methods for managing uncertainty whenever it wants to leave toy micro-worlds to deal with the real world. Ask not what you can do for the theory of uncertainty; ask what the theory of uncertainty can do for you. Spencer Star Bitnet: star@lavalvm1.bitnet Arpanet: star@b.gp.cs.cmu.edu