Daniel J. Navarro

Rational Approximations to Rational Models: Alternative Algorithms for Category Learning

Rational models of cognition typically consider the abstract computational problems posed by the environment, assuming that people are capable of optimally solving those problems. This differs from more traditional formal models of cognition, which focus on the psychological processes responsible for behavior. A basic challenge for rational models is thus explaining how optimal solutions can be approximated by psychological processes. We outline a general strategy for answering this question, namely to explore the psychological plausibility of approximation algorithms developed in computer science and statistics. In particular, we argue that Monte Carlo methods provide a source of rational process models that connect optimal solutions to psychological processes. We support this argument through a detailed example, applying this approach to Andersons (1990, 1991) rational model of categorization (RMC), which involves a particularly challenging computational problem. Drawing on a connection between the RMC and ideas from nonparametric Bayesian statistics, we propose 2 alternative algorithms for approximate inference in this model. The algorithms we consider include Gibbs sampling, a procedure appropriate when all stimuli are presented simultaneously, and particle filters, which sequentially approximate the posterior distribution with a small number of samples that are updated as new data become available. Applying these algorithms to several existing datasets shows that a particle filter with a single particle provides a good description of human inferences.

Global model analysis by parameter space partitioning

To model behavior, scientists need to know how models behave. This means learning what other behaviors a model can produce besides the one generated by participants in an experiment. This is a difficult problem because of the complexity of psychological models (e.g., their many parameters) and because the behavioral precision of models (e.g., interval-scale performance) often mismatches their testable precision in experiments, where qualitative, ordinal predictions are the norm. Parameter space partitioning is a solution that evaluates model performance at a qualitative level. There exists a partition on the models parameter space that divides it into regions that correspond to each data pattern. Three application examples demonstrate its potential and versatility for studying the global behavior of psychological models.

Assessing the distinguishability of models and the informativeness of data.

A difficulty in the development and testing of psychological models is that they are typically evaluated solely on their ability to fit experimental data, with little consideration given to their ability to fit other possible data patterns. By examining how well model A fits data generated by model B, and vice versa (a technique that we call landscaping), much safer inferences can be made about the meaning of a models fit to data. We demonstrate the landscaping technique using four models of retention and 77 historical data sets, and show how the method can be used to: (1) evaluate the distinguishability of models, (2) evaluate the informativeness of data in distinguishing between models, and (3) suggest new ways to distinguish between models. The generality of the method is demonstrated in two other research areas (information integration and categorization), and its relationship to the important notion of model complexity is discussed.

Better explanations of lexical and semantic cognition using networks derived from continued rather than single word associations

In this article, we describe the most extensive set of word associations collected to date. The database contains over 12,000 cue words for which more than 70,000 participants generated three responses in a multiple-response free association task. The goal of this study was (1) to create a semantic network that covers a large part of the human lexicon, (2) to investigate the implications of a multiple-response procedure by deriving a weighted directed network, and (3) to show how measures of centrality and relatedness derived from this network predict both lexical access in a lexical decision task and semantic relatedness in similarity judgment tasks. First, our results show that the multiple-response procedure results in a more heterogeneous set of responses, which lead to better predictions of lexical access and semantic relatedness than do single-response procedures. Second, the directed nature of the network leads to a decomposition of centrality that primarily depends on the number of incoming links or in-degree of each node, rather than its set size or number of outgoing links. Both studies indicate that adequate representation formats and sufficiently rich data derived from word associations represent a valuable type of information in both lexical and semantic processing.

Common and distinctive features in stimulus similarity: A modified version of the contrast model

Featural representations of similarity data assume that people represent stimuli in terms of a set of discrete properties. In this article, we consider the differences in featural representations that arise from making four different assumptions about how similarity is measured. Three of these similarity models— the common features model, the distinctive features model, and Tversky’s seminal contrast model—have been considered previously. The other model is new and modifies the contrast model by assuming that each individual feature only ever acts as a common or distinctive feature. Each of the four models is tested on previously examined similarity data, relating to kinship terms, and on a new data set, relating to faces. In fitting the models, we have used the geometric complexity criterion to balance the competing demands of data-fit and model complexity. The results show that both common and distinctive features are important for stimulus representation, and we argue that the modified contrast model combines these two components in a more effective and interpretable way than Tversky’s original formulation.

Extending the ALCOVE model of category learning to featural stimulus domains

The ALCOVE model of category learning, despite its considerable success in accounting for human performance across a wide range of empirical tasks, is limited by its reliance on spatial stimulus representations. Some stimulus domains are better suited to featural representation, characterizing stimuli in terms of the presence or absence of discrete features, rather than as points in a multidimensional space. We report on empirical data measuring human categorization performance across a featural stimulus domain and show that ALCOVE is unable to capture fundamental qualitative aspects of this performance. In response, a featural version of the ALCOVE model is developed, replacing the spatial stimulus representations that are usually generated by multidimensional scaling with featural representations generated by additive clustering. We demonstrate that this featural version of ALCOVE is able to capture human performance where the spatial model failed, explaining the difference in terms of the contrasting representational assumptions made by the two approaches. Finally, we discuss ways in which the ALCOVE categorization model might be extended further to use “hybrid” representational structures combining spatial and featural components.

Hypothesis Generation, Sparse Categories, and the Positive Test Strategy

We consider the situation in which a learner must induce the rule that explains an observed set of data but the hypothesis space of possible rules is not explicitly enumerated or identified. The first part of the article demonstrates that as long as hypotheses are sparse (i.e., index less than half of the possible entities in the domain) then a positive test strategy is near optimal. The second part of this article then demonstrates that a preference for sparse hypotheses (a sparsity bias) emerges as a natural consequence of the family resemblance principle; that is, it arises from the requirement that good rules index entities that are more similar to one another than they are to entities that do not satisfy the rule.

Latent features in similarity judgments: A nonparametric bayesian approach

One of the central problems in cognitive science is determining the mental representations that underlie human inferences. Solutions to this problem often rely on the analysis of subjective similarity judgments, on the assumption that recognizing likenesses between people, objects, and events is crucial to everyday inference. One such solution is provided by the additive clustering model, which is widely used to infer the features of a set of stimuli from their similarities, on the assumption that similarity is a weighted linear function of common features. Existing approaches for implementing additive clustering often lack a complete framework for statistical inference, particularly with respect to choosing the number of features. To address these problems, this article develops a fully Bayesian formulation of the additive clustering model, using methods from nonparametric Bayesian statistics to allow the number of features to vary. We use this to explore several approaches to parameter estimation, showing that the nonparametric Bayesian approach provides a straightforward way to obtain estimates of both the number of features and their importance.

A note on the applied use of MDL approximations

An applied problem is discussed in which two nested psychological models of retention are compared using minimum description length (MDL). The standard Fisher information approximation to the normalized maximum likelihood is calculated for these two models, with the result that the full model is assigned a smaller complexity, even for moderately large samples. A geometric interpretation for this behavior is considered, along with its practical implications.

Language Evolution Can Be Shaped by the Structure of the World.

Human languages vary in many ways but also show striking cross-linguistic universals. Why do these universals exist? Recent theoretical results demonstrate that Bayesian learners transmitting language to each other through iterated learning will converge on a distribution of languages that depends only on their prior biases about language and the quantity of data transmitted at each point; the structure of the world being communicated about plays no role (Griffiths & Kalish, 2005, 2007). We revisit these findings and show that when certain assumptions about the relationship between language and the world are abandoned, learners will converge to languages that depend on the structure of the world as well as their prior biases. These theoretical results are supported with a series of experiments showing that when human learners acquire language through iterated learning, the ultimate structure of those languages is shaped by the structure of the meanings to be communicated.