Algebraic aspects of Bayesian modeling in psychology

Several Bayesian models in perceptual and cognitive psychology show a complex organization that is only partly captured by referring to Bayes rule and the associated concepts of prior, likelihood, and posterior distributions. In this article, an algebraic framework is constructed that may serve as a guide for representing and analyzing the internal organization of these models. The construction begins by defining a comprehensive class of probabilistic structures, called probability kernels, and by focusing on three basic operations on them, called projection, conditioning, and promotion. The expressive power of these operations is then illustrated by showing how suitable combinations of them cover typical moves of Bayesian computations, such as mixing or inducing probability distributions, marginalizing likelihood functions, and Bayes rule itself. As a central finding, we show that a suitable combination of the basic operations gives rise to a binary relation that organizes any complete set of probability kernels as a lattice, and we highlight the peculiarities of this algebraic structure. Lastly, we illustrate the analytic use of the proposed framework by applying it to two Bayesian models from the literature, one concerning spatial vision and the other concerning category representation.