How cognitive modeling can benefit from hierarchical Bayesian models
Introduction
Bayesian statistics provides a compelling and influential framework for representing and processing information. Over the last few decades, it has become the major approach in the field of statistics, and has come to be accepted in many or most of the physical, biological and human sciences. This paper, and this special issue, are about what one particular niche within Bayesian statistics, in the form of hierarchical models, can contribute to cognitive modeling.
It would be wrong to claim that there is complete agreement on exactly how Bayesian analyses should be conducted and interpreted. Like any powerful and fundamental idea, it can be conceived and formulated in a variety of ways. At the basic theoretical level, the ‘objective Bayesian’ approach expounded by Jaynes (2003) encourages a different style of thinking about Bayesian analysis than the ‘subjective Bayesian’ approach of de Finetti (1974). At the practical level of conducting Bayesian analyses, there is also a spectrum, ranging from work that closely follows the objective viewpoint (e.g., Gregory, 2005, Sivia, 1996), to work that is more agnostic or adopts a naturally subjective position (e.g., Congdon, 2006, Gelman et al., 2004, Gelman and Hill, 2007). There are many additional subtleties and perspectives in the excellent accounts provided by Bernado and Smith (2000), Lindley (1972) and MacKay (2003) and others.
But Bayesian statistics is in agreement on the very basic issues. Knowledge and uncertainty about variables is represented by probability distributions, and this knowledge can be processed, updated, summarized, and otherwise manipulated using the laws of probability theory. These commitments distinguish Bayesian statistics from other competing frameworks, especially those based on frequentist views of probability, and sampling distribution approaches to handling uncertainty. What Bayesian statistics offers is a remarkably complete, coherent and intuitive method for understanding what is known, based on the assumptions being made, and the information that is available.
Because Bayesian statistics provides a formal framework for making inferences, there are different ways it can be applied in cognitive modeling. One way is to use Bayesian methods as a statistician would, as a method for conducting standard analyses of data. Traditionally, the framework for statistical inference based on sampling distributions and null hypothesis significance testing has been used. Calls for change, noting the clear superiority of Bayesian methods, date back at least to the seminal paper of Edwards, Lindman, and Savage (1963), and have grown more frequent and assertive in the past few years (e.g., Gallistel, 2009, Kruschke, 2010a, Lee and Wagenmakers, 2005, Wagenmakers, 2007). It seems certain Bayesian statistics will play a progressively more central role in the way cognitive science analyzes its data.
A second possibility is to apply Bayesian methods to cognitive modeling as a theoretician would, as a working assumption about how the mind makes inferences. This has been an influential theoretical position for the last decade or so in the cognitive sciences (e.g., Chater et al., 2006, Griffiths et al., 2008). Most existing work has focused on providing ‘rational’ accounts of psychological phenomena, pitched at the computational level within the three-level hierarchy described by Marr (1982). These models generally use Bayesian inference as an account of why people behave as they do, without trying to account for the mechanisms, processes or algorithms that produce the behavior, nor how those processes are implemented in neural hardware. More recently, however, there have also been attempts to apply computational sampling approaches from Bayesian statistics as a theoretical metaphors at the algorithmic and implementation levels. In this work, models are developed in which people mentally sample information (e.g., Sanborn, Griffiths, & Shiffrin, 2010). These uses of Bayesian statistics as theoretical analogies have led to impressive new models, and raised and addressed a range of important theoretical questions. As with all theoretical metaphors–including previous ones like information processing and connectionist metaphors–“Bayes in the head” constitutes a powerful theoretical perspective, but leaves room for other complementary approaches.
A third way to use Bayesian statistics in cognitive science is to use them to relate models of psychological processes to data (e.g., Lee, 2008, Rouder et al., 2005, Wetzels et al., 2010). This is different from the data analysis approach, because the focus is not generic statistical models like the generalized linear model. Instead the goal is relate a detailed model of some aspect of cognition to behavioral or other observed data. One way to think of the distinction is that data analysis typically does inference on the measured dependent variables from an experimental design–measures of recall, learning, response times, and so on–whereas modeling applications typically do inference on latent psychological parameters–memory capacities, learning rates, decision criteria, and so on–that control the behavioral predictions of the model. It is also different from the use of Bayesian inference as a metaphor for the mind (Kruschke, 2010b). There is no requirement that the cognitive models being related to data make Bayesian assumptions. Instead, they are free to make any sort of processing claims about how cognition works. The goal is simply to use Bayesian statistical methods to evaluate the proposed model against available data.
This third approach is the focus of the current special issue. We think it is an especially interesting, important, and promising approach, precisely because it deals with fully developed models of cognition, without constraints on the theoretical assumptions used to develop the models. The idea is to begin with existing theoretically grounded and empirically successful models of cognition, and embed them within a hierarchical Bayesian framework. This embedding opens a vista of potential extensions and improvements to current modeling, because it provides a capability to model the rich structure of cognition in complicated settings.
In the remainder of this paper, we identify four major new capabilities offered by the hierarchical Bayesian extensions of cognitive models. We discuss each capability, focusing on how it can help theory and model development, and identifying places where they have already been applied, or could and should be applied soon.
Section snippets
Benefits of using hierarchical Bayes in cognitive modeling
Before discussing the potential contribution hierarchical Bayesian methods can make to cognitive modeling, we need to say what we mean by ‘hierarchical Bayes’. We do that in the next section–by characterizing its complement, in the form of the currently dominant non-hierarchical modeling approach–and then discuss the advantages of the hierarchical approach.
Conclusion
Non-hierarchical approaches to understanding cognitive processes dominate the current landscape. The basic approach can probably fairly be caricatured as one of identifying a psychological phenomenon (e.g., generalization, memory, decision-making), finding an interesting task relating to some aspect of the phenomenon (e.g., similarity judgments, recall, two-alternative forced-choice decisions) and building a model that can fit empirical data from the task using a few psychologically meaningful
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