Patterns, Types, and Bayesian Learning

Consider a probability distribution governing the evolution of a descrete-time stochastic process. Such a distribution may be represented as a convex combination of more elementary probability measures, with the interpretation of a two-stage Bayesian procedure. In the first stage, one of the measures is randomly selected according to the weights of the convex combinations (i.e., their prior probabilities), and in the second stage the selected measure governs the evolution of the stochastic process. Generally, however, the original distribution has infinitely many different insights about the process depending on the representation with which they start. This paper identifies one endogenous representation which is natural in the sense that its component measures are precisely the learnable probabilistic patterns.