Sampling Assumptions and the Size Principle in Property Induction

Sampling Assumptions and the Size Principle in Property Induction Philip M. Fernbach (philip_fernbach@brown.edu) Brown University Department of Cognitive and Linguistic Sciences, Box 1978 Providence, RI 02912 USA Abstract The ‘size principle’ emphasized in recent Bayesian models of inductive generalization (Kemp & Tenenbaum, 2003; Sanjana & Tenenbaum, 2003; Tenenbaum & Griffiths, 2001) is tested in the domain of property induction. As predicted by the model, the size principle is obeyed more frequently when a strong sampling assumption is explicit than when sampling is weak or unspecified, but it is not followed consistently. This implies that, although people are sometimes sensitive to sampling assumptions as specified by the Bayesian framework, models based on a strong sampling assumption may not provide general accounts of property induction. Introduction From the time of their introduction in the 16 th century until general acceptance in the 18 th century, tomatoes were not widely eaten in Mediterranean Europe. Due to their morphological resemblance to the nightshade plant, they were assumed poisonous and avoided. This is a striking — if unsuccessful — example of a common strategy that human beings utilize to make sense of the world: to induce possession of a property by one category from another. Human beings perform these inductions naturally, often from few examples and even between categories that are prima facie dissimilar. Understanding the nature of this type of inference is a major project of cognitive science. Empirical studies of property induction have been conducted since the mid 1970s using argument strength ratings for premise-conclusion pairs and blank predicates. Figure 1 provides an example. Salmon have sesamoid bones Lions have sesamoid bones Therefore Polar Bears have sesamoid bones Figure 1: Property induction argument A host of phenomena have been observed concerning how people evaluate the strength of such arguments (reviewed by Sloman & Lagnado, 2005). Several models have been proposed to account for the phenomena. The models fall into two camps and as such are revealing of a deeper methodological divergence in cognitive science. On the one hand are models directly motivated by empirical work and aimed at the level of algorithm (Marr, 1982). The two best known were proposed in the 1990s (Osherson et al., 1990; Sloman, 1993) and rely on notions of similarity and feature matching. These models tend to be good explanatory accounts of the phenomena that have been studied but their scope is limited. On the other hand are models which come from the tradition of rational analysis (Anderson, 1990). These models seek to identify the computational problem the system is trying to solve and propose an optimal solution. A common theme of the models of this type has been that they utilize the tools of Bayesian statistics. Heit (1998) was the first to apply the Bayesian framework to property induction, providing an account of several of the empirical phenomena. More recently a family of models has emerged that has advanced the project by introducing a principled likelihood calculation based on sampling assumptions, proposing a general method for generating a hypothesis space and attempting to ground the prior distribution in domain specific theories (Kemp & Tenenbaum, 2003; Tenenbaum & Griffiths, 2001; Sanjana & Tenenbaum, 2003). I refer to this family of models collectively as the Sampling Sensitive Bayesian (SSB) model. In this paper I evaluate the approaches by testing a key prediction of the SSB model. The SSB model relies on a likelihood calculation that is dependent on assumptions about how the data are sampled. This implies that argument strength judgments should be sensitive to variations in sampling procedures. Specifically, the SSB model generally assumes that human property induction is best described using strong sampling (Kemp & Tenenbaum, 2003; Sanjana & Tenenbaum, 2000) an assumption that examples are drawn at random from the set of objects to which the predicate applies. This assumption leads to a ‘size principle,’ a preference for smaller hypotheses over larger ones given data that is consistent with both. If the size principle holds, then given more examples within a range, participants should be less willing to generalize outside of that range, a tendency that should manifest itself as an inductive non-monotonicity whereby adding similar premises decreases property induction to a dissimilar conclusion. This phenomenon is not predicted by either Osherson et al’s (1990) Similarity/Coverage (SC) model or Sloman’s (1993) Feature-Based (FB) model as neither of these models is sensitive to variations in sampling procedures. To identify the source of this divergence I first present the similarity and feature-based models. I then present the SSB model and show how the size principle emerges from the model and why it does not apply to other models. Lastly I present an argument preference experiment designed at testing for the size principle and discuss the implications of the results.