Desirable properties of Bayesian models in perceptual psychology. A discussion of distribution families closed under multiplication

Abstract

One reason why Bayesian models based on Normal distributions are welcome in perceptual psychology is that they fit three natural expectations in the area: when two stimulus factors are in conflict the resulting perceptual value lies between the values separately supported by those factors, the resulting value is closer to that supported by the more reliable factor, and the variance implied by the combined action of the factors is no greater than the variances implied by them separately. To what degree are these good qualities shared by Bayesian models based on probability distributions other than the Normal ones? We address this problem by referring to distribution families that satisfy closure under multiplication, which is a key characteristic of prior families in Bayesian models, and by distinguishing seven desirable properties, one concerning the mean, three the variance, and three the relationship between mean and variance. The distribution families we discuss are the Bernoulli, Geometric, Gamma (including Exponential), Beta, and Pareto families. Through propositions, counterexamples, and graphical demonstrations, we show that these families vary in the degree to which they approximate the optimum performance of the Normal. We conclude by noting the relevance of our results for the Bayesian concept of information integration in psychology.