Hierarchical Bayesian Models

Those of us who study human cognition have no easy task. We try to understand how people functionally represent and processes information in performing cognitive activities such as vision, perception, memory, language, and decision making. Fortunately, experimental psychology has a rich theoretical tradition, and there is no shortage of insightful theoretical proposals. Also, it has a rich experimental tradition with a multitude of experimental techniques for isolating purported processes. What it lacks, however, is a rich statistical tradition to link theory to data. At the heart of the field is the difficult task of trying to use data from experiments to inform theory, that is, to understand accurately the relationships within the data and how they provide evidence for or against various theoretical positions. The difficulty in linking data to theory can be seen in a classic example from Estes (1956). Estes considered two different theories of learning: one in which learning was gradual and another where learning happened all at once. These two accounts are shown in Figure 1A. Because these accounts are so different, adjudicating between them should be trivial: one simply examines the data for either a step function or a gradual change. Yet, in many cases, this task is surprisingly difficult. To see this difficulty, consider the data of Ritter and Reder (1992), who studied the speed up in response times from repeated practice of a mathematics tasks. The data are shown in Figure 1B, and the grey lines show the data from individuals. These individual data are highly variable making it impossible to spot trends. A first-order approach is to simply take the means across individuals at different levels of practice, and these means (red points) decrease gradually, seemingly providing support for the gradual theory of learning. Estes, however, noted that this pattern does not necessarily imply that learning is gradual. Instead, learning might be all-at-once, but the time at which different individuals transition may be different. Figure 1C shows an example; for demonstration purposes, hypothetical data are shown without noise. If data are generated from the all-at-once model and there is variation in this transition time, then the mean will reflect the proportion of individuals in the unlearned state at a given level of practice. This proportion may decrease