Multidimensional Models of Perception and Cognition

F. Gregory Ashby

• Publisher: Psychology Press
• Year: 2014
• eBook ISBN: 9781317784036
• Topic: Education
• Subtopic: Teaching Mathematics

The mental representations of perceptual and cognitive stimuli vary on many dimensions. In addition, because of quantal fluctuations in the stimulus, spontaneous neural activity, and fluctuations in arousal and attentiveness, mental events are characterized by an inherent variability. During the last several years, a number of models and theories have been developed that explicitly assume the appropriate mental representation is both multidimensional and probabilistic. This new approach has the potential to revolutionize the study of perception and cognition in the same way that signal detection theory revolutionized the study of psychophysics. This unique volume is the first to critically survey this important new area of research.

1 Multivariate Probability Distributions

F. Gregory Ashby

University of California at Santa Barbara

Many of the models discussed in this book are based on the assumption that the perceptual effect of a stimulus is random over trials, although on any single trial is has a specified fixed value. This assumption, which can be traced back to Fechner (1860, 1966), was fully exploited in signal detection theory (e.g., Green & Swets, 1974) where the focus was on unidimensional perceptual representations. The models in this book focus on multidimensional representations. Although the mathematical basis of these models is probability theory, the generalization from univariate to multivariate probability distributions involves several complications. This first chapter reviews many of the important results of multivariate probability theory upon which the later chapters depend.

We assume that most readers will have some familiarity with univariate probability theory and with the basics of matrix algebra. The first section in this chapter is a very brief survey of univariate probability theory, and those readers unfamiliar with this material might wish to supplement it with readings from an outside source, such as Parzen (1960). This chapter does not contain a review of the basics of matrix algebra, so those readers unfamiliar with basic matrix operations such as addition, multiplication, transposition, and inversion should consult any introductory matrix algebra text (often called linear algebra; e.g., Noble & Daniel, 1977).

Readers familiar with multivariate probability theory might still wish to skim this chapter. Several sections are included that contain some useful but little known results. These include techniques for generating random samples from multivariate normal distributions, for quickly performing certain numerical integrations, and for computing the predicted accuracy of the ideal observer in multidimensional categorization and identification experiments.

UNIVARIATE PROBABILITY THEORY

When there is only one stimulus dimension, many of the models in this book become equivalent to signal detection theory (e.g., Green & Swets, 1974). As an example of a signal detection application, consider a two-alternative forced-choice task in which the stimulus ensemble contains two stimuli differing only in intensity. Call the two intensities A₁ and A₂, and denote the perceived intensity, when a single stimulus is presented, by X. The fundamental assumption of signal detection theory is that X is a random variable.

One way to characterize the subject’s perceptual experience is to determine the probability that the perceived intensity falls in certain intervals. To aid these calculations, we use certain standard mathematical functions. The cumulative probability distribution function F_i(x) gives the probability that the perceived intensity is less than or equal to a specified value x on trials when the intensity is A_i. Formally,

The cumulative distribution function increases monotonically from 0 to 1.0 as x increases. The second standard function, known as the probability density function, is defined as the derivative of F_i(x). Specifically,

Whereas values of the cumulative distribution function are themselves probabilities, values of the density function are called likelihoods, and are not probabilities. Given the density function, probabilities can be found by integration. For example,

and so probabilities are areas under the probability density function. From Equation 2 it is clear that

To see that f_i(x) is not a probability, note that even if f_i(x) > 0 for some value of x, if f_i(x) is continuous then it is always true that

Given the probability density function of a random variable X, the mean of X, denoted μ_x, is defined as the expected value of X, E (X), and can be computed via

The variance of X is defined as its expected squared deviation from the mean:

In general, suppose Y = h(X) is any function of the random variable X and that we are interested in the expected value of Y. Equation 3 indicates that this requires knowledge of the ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• PREFACE
• 1 MULTIVARIATE PROBABILITY DISTRIBUTIONS
• PART I: SIMILARITY, PREFERENCE, AND CHOICE
• PART II: INTERACTIONS BETWEEN PERCEPTUAL DIMENSIONS
• PART III: DETECTION, IDENTIFICATION, AND CATEGORIZATION
• References
• Author Index
• Subject Index