Chapter 11: From Statistics to Beliefs

Overview

"In fact, all the complex mass of statistics produced by the sports industry can without exception be produced not only more economically by computer, but also with more significant patterns and more amazing freaks. I take it the main object of organized sport is to produce a profusion of statistics?"

"Oh, yes", said Rowe. "So far as I know."

—Michael Frayn, The Tin Men

Section 10.3 shows that, for first-order reasoning about probability, it is possible to put a probability both on the domain and on the set of possible worlds. Putting a probability on the domain is appropriate for "statistical" reasoning, while putting a probability on the set of possible worlds can be viewed as capturing an agent's subjective beliefs. Clearly the two should, in general, be related. That is, if an agent's knowledge base includes statistical information, his subjective probabilities should reflect this information appropriately. Relating the two is quite important in practice. Section 1.1 already has an example of this. Recall that, in this example, a doctor with a patient Eric can see that Eric has jaundice, no temperature, and red hair. His medical textbook includes the statistical information that 90 percent of people with jaundice have hepatitis and 80 percent of people with hepatitis have a temperature. What should the doctor's degree of belief be that Eric has hepatitis? This degree of belief is important because it forms the basis of the doctor's future decision regarding the course of treatment.

Unfortunately, there is no definitive "right" way for relating statistical information to degrees of belief. In this chapter, I consider one approach for doing this that has some remarkable properties (unfortunately, not all of them good). It is closely related to maximum entropy (at least, in the case of first-order language with only unary predicates) and gives insight into default reasoning as well. For definiteness, I focus on probabilistic reasoning in this chapter. Many of the ideas presented here should be applicable to other representations of uncertainty, but to date there has been no work on this topic. I also assume for simplicity that there is only one agent in the picture.