Psychophysical functions

Psychophysics is concerned with measuring how external physical stimuli cause internal psychological sensations. In a typical psychophysical experiment, subjects are repeatedly confronted with two similar stimuli, such as two sounds, two weights, two smells, two lines, or two time intervals. One stimulus—the standard stimulus—always has the same intensity, whereas the other stimulus—the test stimulus—varies in intensity from trial to trial. On each trial of the experiment, the subject's task is to detect which of the two stimuli is more intense: louder, heavier, stronger-smelling, more tilted, or longer lasting. The more similar the stimuli, the more difficult it is for the subject to discriminate between them.

The relation between task difficulty and performance usually follows a sigmoid or S-shaped curve, as shown in Figure 12.1. The x-axis represents differences in stimulus intensity between the test stimulus and the standard. The y-axis represents the probability of a response indicating that the test stimulus has higher intensity. The curve linking these physical and psychological measures is known as a psychophysical function, and is used to define several values of interest. The “point of subjective equality” (PSE) represents that difference in intensity for which the participant chooses the correct response 50% of the time, which is not necessarily the point where the two stimuli are equally intense physically. The “just noticeable difference” (JND) is the intensity threshold at which the subject “just” notices a difference in intensity between two stimuli.

Take-the-best

The take-the-best (TTB) model of decision-making (Gigerenzer & Goldstein, 1996) is a simple but influential account of how people choose between two stimuli on some criterion, and a good example of the general class of heuristic decision-making models (e.g., Gigerenzer & Todd, 1999; Gigerenzer & Gaissmaier, 2011; Payne, Bettman, & Johnson, 1990). TTB addresses decision tasks like “which of Frankfurt or Munich has the larger population?”, “which of a catfish and a herring is more fertile?”, and “which of these two professors has the higher salary?”.

TTB assumes that all stimuli are represented in terms of the presence or absence of a common set of cues. In the well-studied German cities data set, this means cities are represented in terms of nine cues, including whether or not they have an international airport, whether they have hosted the Olympics, and whether they have a football team in the Bundesliga. Associated with each cue in TTB is a “cue validity.” This validity measures the proportion of times that, for those pairs of stimuli where one has the cue and the other does not, the cue belongs to the stimulus that has the greater criterion value. For example, the cue “Is the city the national capital?” is highly valid because the capital city, Berlin, is also the most populous city.

The TTB model assumes that, when people decide which is the larger of two cities, they search the cues from highest to lowest validity, stopping as soon as a cue is found that one city has but the other does not. At this point, TTB says simply that people choose the city that has the cue. If all of the cues are exhausted, TTB assumes people guess.

The SIMPLE model

Brown, Neath, and Chater (2007) proposed the SIMPLE (Scale-Invariant Memory, Perception, and LEarning) model, which, among various applications, has been applied to the basic memory phenomenon of free recall. In this application, the SIMPLE model assumes memories are encoded by the time they were presented, but that the representations are logarithmically compressed, so that more temporally distant memories are more similar. It also assumes that distinctiveness plays a central role in performance on memory tasks, and that interference rather than decay is responsible for forgetting. Perhaps most importantly, the SIMPLE model assumes that the same memory processes operate at all time scales, unlike theories and models that assume different mechanisms for short-term and long-term memory.

The first application considered by Brown et al. (2007) involves seminal immediate free recall data reported by Murdock (1962). The data give the proportion of words correctly recalled averaged across participants, for lists of 10, 15, and 20 words presented at a rate of 2 seconds per word, and lists of 20, 30, and 40 words presented at a rate of 1 second per word.

Brown et al. (2007) make some reasonable assumptions about undocumented aspects of the task (e.g., the mean time of recall from the end-of-list presentation), to set the time Ti between learning and retrieval of the ith item. With these times established, the application of the SIMPLE model to the free recall data involves five stages, which are clearly described in Brown et al. (2007, Appendix).

This book, together with the code, answers to questions, and other material at www.bayesmodels.com, teaches you how to do Bayesian modeling. Using modern computer software—and, in particular, the WinBUGS program—this turns out to be surprisingly straightforward. After working through the examples provided in this book, you should be able to build your own models, apply them to your own data, and draw your own conclusions.

This book is based on three principles. The first is that of accessibility: the book's only prerequisite is that you know how to operate a computer; you do not need any advanced knowledge of statistics or mathematics. The second principle is that of applicability: the examples in this book are meant to illustrate how Bayesian modeling can be useful for problems that people in cognitive science care about. The third principle is that of practicality: this book offers a hands-on, “just do it” approach that we feel keeps students interested and motivated.

In line with these three principles, this book has little content that is purely theoretical. Hence, you will not learn from this book why the Bayesian philosophy to inference is as compelling as it is; neither will you learn much about the intricate details of modern sampling algorithms such as Markov chain Monte Carlo, even though this book could not exist without them.