11.1 Reference Classes

Before going into the technical details of the approach, it is worth examining in more detail some properties that have been considered desirable for a method for going from statistical information to degrees of belief. This is perhaps best done by considering the traditional approach to the problem, which uses what are called reference classes.To simplify matters, assume for the purposes of this discussion that the agent's knowledge base consists of two types of statements: statistical assertions of the form "90 percent of people with jaundice have hepatitis" and "80 percent of people with hepatitis have a temperature" and information about one individual (such as Eric). The problem is to determine appropriate degrees of belief regarding events concerning that individual, given the statistical information and the information about the individual.

The idea of the reference-class approach is to equate the degree of belief in propositions about an individual with the statistics from a suitably chosen reference class (i.e., a set of domain individuals that includes the individual in question) about which statistics are known. For example, if the doctor is interested in ascribing a degree of belief to the proposition "Eric has hepatitis", he would first try to find the most suitable reference class for which he has statistics. Since all the doctor knows about Eric is that Eric has jaundice, then the set of people with jaundice seems like a reasonable reference class to use. Intuitively, the reference class is a set of individuals of which Eric is a "typical member." To the extent that this is true, then Eric ought to be just as likely to satisfy a property as any other member of the reference class. Since someone chosen at random from the set of people with jaundice has probability .9 of having hepatitis, the doctor assigns a degree of belief of .9 to Eric's having hepatitis.

While this seems like a reasonable approach (and not far from what people seem to do in similar cases), it is often difficult to apply in practice. For example, what if the doctor also knows that Eric is a baby and only 10 percent of babies with jaundice have hepatitis. What reference class should he use in that case? More generally, what should be done if there are competing reference classes? And what counts as a legitimate reference class?

To understand these issues, consider the following examples. To start with, consider the situation where Eric is a baby and only 10 percent of babies with jaundice have hepatitis. In this case, the standard response is that the doctor should prefer the more specific reference class—technically, this means the doctor should use the smallest reference class for which he has statistics. Since the set of babies is a subset of the set of people, this heuristic suggests the doctor ascribe degree of belief .1 to Eric's having hepatitis, rather than .9.

But the preference for the more specific reference class must be taken with a grain of salt, as the following example shows:

Example 11.1.1

Consider again the first knowledge base, where the doctor does not know that Eric is a baby. In that case, it seems reasonable for the doctor to take the appropriate reference class to consist of all people with jaundice and ascribe degree of belief .9 to Eric's having hepatitis. But Eric is also a member of the reference class consisting of jaundiced patients without hepatitis together with Eric. If there are quite a few jaundiced patients without hepatitis (e.g., babies), then there are excellent statistics for the proportion of patients in this class with hepatitis: it is approximately 0 percent. Eric is the only individual in the class who may have hepatitis! Moreover, this reference class is clearly more specific (i.e., a subset of) the reference class of all people with jaundice. Thus, a naive preference for the more specific reference class results in the doctor ascribing degree of belief 0 (or less than ∊ for some very small ∊) to Eric's having hepatitis! Clearly there is something fishy about considering the reference class consisting of jaundiced patients that do not have hepatitis together with Eric, but exactly what makes this reference class so fishy?

There are other problems with the reference-class approach. Suppose that the doctor also knows that Eric has red hair but has no statistics for the fraction of jaundiced people with red hair who have hepatitis. Intuitively, the right thing to do in this case is ignore the fact that Eric has red hair and continue to ascribe degree of belief .9 to Eric's having hepatitis. Essentially, this means treating having red hair as irrelevant. But what justifies this? Clearly not all information about Eric is irrelevant; for example, discovering that Eric is a baby is quite relevant.

This discussion of irrelevance should seem reminiscent of the discussion of irrelevance in the context of default reasoning (Section 8.5). This is not an accident. It turns out the issues that arise when trying to ascribe degrees of belief based on statistical information are much the same as those that arise in default reasoning. This issue is discussed in more detail in Section 11.4.

Going back to Eric, while it seems reasonable to prefer the more specific reference class (assuming that the problems of deciding what counts as a reasonable reference class can be solved), what should the doctor do if he has two competing reference classes? For example, suppose that the doctor knows that 10 percent of babies with jaundice have hepatitis but 90 percent of Caucasians with jaundice have hepatitis, and that Eric is a Caucasian baby with jaundice. Now the doctor has two competing reference classes: Caucasians and babies. Neither is more specific than the other. In this case, it seems reasonable to somehow weight the 10 percent and 90 percent, but how? The reference-class approach is silent on that issue. More precisely, its goal is to discover a single most appropriate reference class and use the statistics for that reference class to determine the degree of belief. If there is no single most appropriate reference class, it does not attempt to ascribe degrees of belief at all.

The random-worlds approach that I am about to present makes no attempt to identify a single relevant reference class. Nevertheless, it agrees with the reference-class approach when there is an obviously "most-appropriate" reference class. Moreover, it continues to make sense even when no reference class stands out as being the obviously most appropriate one to choose.