Finding hidden types: Inductive inference in long-tailed environments

Finding hidden types: Inductive inference in long-tailed environments Daniel J. Navarro (daniel.navarro@adelaide.edu.au) School of Psychology, University of Adelaide, SA 5005, Australia Abstract rather different character than the usual exemplar frequency effects. The key difference is that the effect does not pertain to a specific exemplar, but instead is an effect that pertains to the overall frequency distribution. In the first case, the learner has evidence that the frequency distribution is homogeneous: the observed exemplars have equal frequency. In the second case, the evidence implies that the frequency distribution is long-tailed, meaning that there are a small number of items that are very common, but most observations are quite rare. Making inference in everyday life often requires people to make inferences about low frequency events. In the most ex- treme case, some types of object or event may have never been previously observed. An experiment is presented in which par- ticipants needed to infer the existence and number of unob- served event types, based solely on the frequency distribution of a set of observed events. Results indicate people’s infer- ences are sensitive to the shape of the distribution over the ob- served events, even when the number of observed events and event types is held constant, and that people are able to infer abstract rules that describe entire classes of event distributions. Human inferences are shown to be similar to those made by a hierarchical Bayesian model. Frequency effects in categorization and choice Keywords: inductive inference, Bayesian cognition, fre- quency effects, concept learning Imagine you are walking through the bushlands in a foreign land. You are accompanied by a local guide, who comments on the plant life around you. So far she has described 20 plants as alba, 20 plants as glabra and another 20 as eburnia. On this basis it is tempting to think that albas, glabras and eburnias are the only types of plants around, or at least the only plant types that your guide is intending to label for you. You could not be certain that this is the correct inference of course, but it seems sensible. Contrast this with a slightly different scenario, in which your guide refers to 58 of the plants as albas, points to one example of a glabra and one example of an eburnia. Again, it is impossible to be sure what to believe, but it seems much less reasonable to conclude that these are the only three plant labels that your guide is ever going to use. Both scenarios involve 60 plants and 3 category labels, yet they do not feel equivalent. The logic behind this intuition is relatively straightforward. In the second example, you have evidence of the existence of low-frequency types, whereas in the first example you do not. The fact that some types are relatively rare suggests that there may be other rare types that you have not yet seen. In other words, the shape of the frequency distribution plays a power- ful role in shaping our inductive inferences in this problem. This is illustrated in Figure 1. In essence, this is a category learning problem: the learner has encountered a new kind of object (the foreign plants) and is attempting to learn the extension of the category with re- spect to a particular feature (the labels). Viewed as a cate- gory learning problem, the different inferences drawn in the two cases are an example of a frequency effect, though of a Exemplar frequency effects are well-established in the cate- gorization literature: for instance, high-frequency exemplars are classified more accurately, and are judged to be more typ- ical of the category than are low-frequency items (Nosofsky, 1988). However, although the role of item frequency is well- studied (Nosofsky, 1988; Barsalou, 1985; Barsalou, Hutten- locher, & Lamberts, 1998), the inductive inference described earlier is rather different to exemplar frequency effects as they are traditionally conceived. In both examples the ob- served frequency of blue, purple, white or any other color flower is zero, yet they differ in terms of the expected sub- jective frequency. That is, changing the distribution of the same set of three types (albas, glabras, eburnias) alters the expectations about the probabilities associated with as-yet- unobserved types. Frequency effects of a different kind arise in the judgment and decision making literature. In this literature the focus is on how much weight people place on low-frequency out- comes when evaluating possible options, whereas the concept learning literature tends to focus on the role of high-frequency items. Although much of the early evidence (Kahneman & Tversky, 1979; Tversky & Fox, 1995) suggested that people tend to overweight low-frequency events, there is some evi- dence indicating that this applies primarily to described fre- quencies, and not to experienced ones (e.g. Barron & Erev, 2003; Hertwig, Barron, Weber, & Erev, 2004), though much of this difference can be attributed to the different information and feedback available to participants (e.g. Rakow, Demes, & Newell, 2008; Camilleri & Newell, 2011). As with the cate- gory learning literature, these studies have focused on events whose observed frequency is at least one, rather than looking at the inferences people make about never-observed events.