Merging with a set of probability measures: A characterization

Abstract

In this paper, I provide a characterization of a set of probability measures with which a prior weakly merges. In this regard, I introduce the concept of condi-tioning rules that represent the regularities of probability measures and define the eventual generation of probability measures by a family of conditioning rules. I then show that a set of probability measures is learnable (i.e., all probability mea-sures in the set are weakly merged by a prior) if and only if all probability mea-sures in the set are eventually generated by a countable family of conditioning rules. I also demonstrate that quite similar results are obtained with almost weak merging. In addition, I argue that my characterization result can be extended to the case of infinitely repeated games and has some interesting applications with regard to the impossibility result in Nachbar (1997, 2005).