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.