Merging with a set of probability measures: a characterization - Yuichi Noguchi (Theoretical Economics, 2015)
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 ``conditioning 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 measures in the set are weakly merged by a prior) if and only if all probability measures 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).
Keywords: C72; C73; D83; 330; Bayesian learning; weak merging; conditioning rules; eventual generation; frequency-based prior