Theoretical Economics 10 (2015), 411–444

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### Merging with a set of probability measures: a characterization

*Yuichi Noguchi*

#### Abstract

In this paper, I provide a characterization of a \textit{set} of probability
measures with which a prior ``weakly merges.'' In this regard, I introduce
the concept of ``conditioning rules'' that represent the \textit{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 \textit{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: Bayesian learning, weak merging, conditioning rules, eventual generation, frequency-based prior

JEL classification: C72, C73, D83

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