The Rational Basis of Representatives - eScholarship
The Rational Basis of Representativeness Joshua B. Tenenbaum & Thomas L. Griffiths Department of Psychology Stanford University Stanford, CA 94305-2130 USA jbt,gruffydd @psych.stanford.edu Abstract Representativeness is a central explanatory construct in cognitive science but suffers from the lack of a principled theoretical account. Here we present a formal definition of one sense of representativeness – what it means to be a good example of a process or category in the context of Bayesian inference. This analysis clarifies the rela- tion between representativeness as an intuitive statistical heuristic and normative principles of inductive inference. It also leads to strong quantitative predictions about peo- ple’s judgments, which compare favorably to alternative accounts based on likelihood or similarity when evaluated on data from two experiments. Why do people think that Linda, the politically ac- tive, single, outspoken, and very bright 31-year-old, is more likely to be a feminist bankteller than to be a bank- teller, even though this is logically impossible? Why do we think that the sequence is more likely than the sequence to be produced by flipping a fair coin, even though both are equally likely? The standard answer in cognitive psychology (Kahneman & Tversky, 1972) is that our brains are designed to judge “represen- tativeness”, not probability: Linda is more representative of feminist banktellers than of banktellers, and is more representative of flipping a fair coin than is despite anything that probability theory tells us. Not only errors in probabilistic reasoning, but numer- ous other phenomena of categorization, comparison, and inference have been attributed to the influence of repre- sentativeness (or prototypicality or “goodness of exam- ple”; Mervis & Rosch, 1981; Osherson, Smith, Wilkie, Lopez, & Shafir, 1990; Rips, 1975). However, a princi- pled account of representativeness has not been easy to come by. Its leading proponents (Kahneman & Tversky, 1996; Mervis & Rosch, 1981) have asserted that rep- resentativeness should be defined only operationally in terms of people’s judgments; an a priori, analytic defini- tion need not be given. Critics have countered that this concept is too vague to serve as an explanation of intu- itive probability judgment (Gigerenzer, 1996). This paper presents a framework for constructing ra- tional models of representativeness, based on a Bayesian analysis of what makes an observation a good example of a category or process. The goal is to identify pre- cisely one sense of representativeness and show that it has a rational basis in normative principles of inductive reasoning. We will first point out some shortcomings of previous accounts based on likelihood or similarity, and show how a Bayesian approach can overcome those problems. We will then compare the quantitative predic- tions of Bayesian, likelihood, and similarity models on two sets of representativeness judgments. Previous approaches Likelihood. In trying to relate intuitions about repre- sentativeness to rational statistical inferences, a natural starting point is the concept of likelihood. Let d denote some observed data, such as a sequence of coin tosses, and h denote some hypothesis about the source of d, such as flipping a fair coin. The probability of observ- ing d given that h is true, P(d h), is called a likelihood. Let R(d, h) denote representativeness – how representa- tive the observation d is of the generative process in h. Gigerenzer & Hoffrage (1995) have proposed that rep- resentativeness, to the extent that it can be defined rig- orously, is equivalent to likelihood: R(d, h) = P(d h). This proposal is appealing in that, other factors aside, the more frequently h leads to observing d, the more repre- sentative d should be of h. It is also consistent with some classic errors in probability judgment, such as the con- junction fallacy: a person is almost certainly more likely to match Linda’s description given that she is a bankteller and a feminist than given only that she is a bankteller. While likelihood and representativeness seem related, however, they are not equivalent. Two observations with equal likelihood may differ in representativeness. Know- ing that and are equally likely to be pro- duced by a fair coin does not change our judgment that the latter is the more representative outcome. Tversky & Kahneman (1983) provide several examples of cases in which a more representative outcome is actually less likely. Any sequence of fair coin flips, such as is less likely than one of its subseqences, such as or , but may easily be more representative. More color- fully, “being divorced four times” is more representative of Hollywood actresses than is “voting democratic”, but the former is certainly less likely. Figure 1 illustrates a simple version of the dissoci- ation between representativeness and likelihood. Each panel shows a sample of three points from a Gaussian distribution. With independent sampling, the total likeli- hood of a sample equals the product of the likelihoods for