Academic journal article Psychonomic Bulletin & Review

Three Case Studies in the Bayesian Analysis of Cognitive Models

Academic journal article Psychonomic Bulletin & Review

Three Case Studies in the Bayesian Analysis of Cognitive Models

Article excerpt

Bayesian statistical inference offers a principled and comprehensive approach for relating psychological models to data. This article presents Bayesian analyses of three influential psychological models: multidimensional scaling models of stimulus representation, the generalized context model of category learning, and a signal detection theory model of decision making. In each case, the model is recast as a probabilistic graphical model and is evaluated in relation to a previously considered data set. In each case, it is shown that Bayesian inference is able to provide answers to important theoretical and empirical questions easily and coherently. The generality of the Bayesian approach and its potential for the understanding of models and data in psychology are discussed.

Psychology as an empirical science progresses through the development of formal models incorporating theoretical ideas designed to explain and predict observations of psychological phenomena. This means that progress in psychology relies upon the quality and completeness of the methods it uses to relate models and data. There is little point in developing theories and models, on the one hand, and collecting data in the laboratory or the field, on the other, if the two cannot be brought into contact in useful ways.

In most empirical sciences, Bayesian methods have been or are rapidly being adopted as the most complete and coherent available way to relate models and data. Psychology has long been aware of problems with traditional frequentist and null hypothesis significance-testing approaches to parameter estimation and model selection, and recognition of the Bayesian alternative has followed from a number of recent articles and special volumes addressing the general issues (e.g., Lee & Wagenmakers, 2005; Myung, Forster, & Browne, 2000; Myung & Pitt, 1997; Pitt, Myung, & Zhang, 2002). Beyond the illustrative applications provided in these general treatments, however, there are few worked examples of Bayesian methods being applied to models at the forefront of modern psychological theorizing. Perhaps one reason is that there has been too great a focus on model selection defined in a narrow sense-particularly through the evaluation of Bayes factors-rather than a full Bayesian analysis. The perception that all that Bayesian methods have to offer for the evaluation of psychological models is a number that quantifies how much more likely one model is than another is dangerously limiting.

In this article, three previous cognitive-modeling studies are revisited, in an attempt to demonstrate the generality and usefulness of the Bayesian approach. The three applications involve the multidimensional scaling (MDS) representation of stimulus similarity (Shepard, 1962, 1980), the generalized context model (GCM) account of category learning (Nosofsky, 1984, 1986), and a signal detection theory (SDT) account of inductive and deductive reasoning (Heit & Rotello, 2005). These applications were chosen in order to span a range of cognitive phenomena, to involve well-known and influential theories, and to put a focus on the ability of Bayesian methods to provide useful answers to important theoretical and empirical questions.

METRIC MULTIDIMENSIONAL SCALING

Theoretical Background

MDS representations of stimuli use a low-dimensional metric space in which points correspond to stimuli and the distance between points models the (dis)similarity between stimuli (Shepard, 1957,1962,1987,1994). Nonmetric varieties of MDS algorithms for inferring these representations from pairwise similarity data (e.g., Kruskal, 1964) make only weak assumptions about the form of the relationship between distance in the MDS space and stimulus similarity. However, Shepard's (1987) universal law of generalization provides a compelling case that similarity decays exponentially with distance, at least for relatively low-level perceptual stimulus domains. …

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