Modeling Individual Differences with Dirichlet Processes
Modeling Individual Differences with Dirichlet Processes Daniel J. Navarro (daniel.navarro@adelaide.edu.au) Department of Psychology, University of Adelaide, SA 5005, Australia Thomas L. Griffiths (thomas griffiths@brown.edu) Department of Cognitive and Linguistic Sciences, Brown University, Providence, RI 02912, USA Mark Steyvers (msteyver@uci.edu) Department of Cognitive Sciences, University of California, Irvine, Irvine CA 92697, USA Michael D. Lee (michael.lee@adelaide.edu.au) Department of Psychology, University of Adelaide, SA 5005, Australia Abstract We introduce a Bayesian framework for modeling indi- vidual differences, in which subjects are assumed to be- long to one of a potentially infinite number of groups. In this model, the groups observed in any particular data set are not viewed as a fixed set that fully explain the variation between individuals, but rather as representa- tives of a latent, arbitrarily rich structure. As more peo- ple are seen, the number of observed groups is allowed to grow, as more details about the individual differences are revealed. We use the Dirichlet process – a distri- bution widely used in nonparametric Bayesian statistics – to define a prior for the model, allowing us to learn flexible parameter distributions without overfitting the data, or requiring the complex computations typically required for determining the dimensionality of a model. As an initial demonstration of the approach, we present an application of the method to categorization data. Much of cognitive science involves the development and evaluation of models. Models formalize theoret- ical predictions and have been successfully applied to a range of phenomena. A recurrent problem, however, is that individual differences between people are often overlooked. This occurs because, most often, models are evaluated against data that have been averaged or aggregated across subjects, and so assume that there are no individual differences between them. In this pa- per we introduce a new framework for modeling individ- ual differences. Informed by recent insights in statistics and machine learning (e.g., Escobar & West, 1995; Neal, 2000), our infinite groups model makes it possible to di- vide subjects who behave similarly into groups, without assuming an upper bound on the number of groups. This model is sufficiently flexible to capture the heterogeneous structure produced by different subjects pursuing differ- ent strategies, allows the number of groups to grow nat- urally as we observe more data, and avoids the complex computations often required for determining the dimen- sionality of an individual differences model. Modeling Individual Differences In those cases of cognitive modeling that recognize indi- vidual differences, it is usually assumed that each subject behaves in accordance with a different parameterization, θ, of a single model, and that model is evaluated against the data from each subject separately (e.g., Ashby, Mad- dox & Lee, 1994; Nosofsky, 1986). Although this avoids the problem of corrupting the underlying pattern of the data, it also foregoes the potential benefits of averag- ing, and guarantees that models are fit to all of the noise in the data. Clearly, individual subject analysis increases the risk of overfitting, and hence reduces the ability to make accurate predictions or to generalize to new contexts. As a result, a number of authors have considered more economical ways of expressing individ- ual differences, which seek to describe the ways in which people are the same as well as the ways in which they are different (e.g., Peruggia, Van Zandt & Chen, 2002; Rouder, Sun, Speckman, Lu & Zhou, in press; Steyvers, Tenenbaum, Wagenmakers & Blum, 2003; Webb & Lee, Two dominant approaches have emerged in the litera- ture on modeling individual differences. In the stochas- tic parameters model (e.g., Peruggia et al., 2002; Rouder et al., in press), every participant is assumed to have a unique parameter value θ that is randomly sampled from a parametric distribution, as illustrated in Figure 1a. In contrast, the groups model assumes that people fall into one of a number of different clusters. Within a group, people are assumed to behave in essentially the same way, but each group is qualitatively different. Under this approach to individual differences modeling (e.g., Lee & Webb, in press; Steyvers et al., 2003; Webb & Lee, 2004), the goal is to partition subjects into a num- ber of groups and associate each group with a parameter set θ, as illustrated by the parameter distribution shown in Figure 1b. Hierarchical Bayesian Models The assumptions underlying these two approaches to modeling individual differences can be understood by viewing both as hierarchical Bayesian models (e.g., Lind- ley & Smith, 1972). If data arise from a parametric dis- tribution x ∼ F (· | θ) described by some cognitive model, then there is assumed to exist a higher-order process G(· | φ) that generates the values of θ. A two level hier- archical model with parameters φ is written, x | θ ∼ G(· | φ) F (· | θ). To apply Bayesian inference to such a model, we also need to define a prior on φ. We will assume that φ ∼ π(·) for an appropriate distribution π(·). In the stochastic parameters model G(· | φ) is usually a tractable distribution such as a Gaussian, with φ corre-