On the Consistency among Prior, Posteriors, and Information Sets (Extended Abstract)

Abstract

This paper studies implications of the consistency conditions among prior, posteriors, and information sets on introspective properties of qualitative belief induced from information sets. The main result reformulates the consistency conditions as: (i) the information sets, without any assumption, almost surely form a partition; and (ii) the posterior at a state is equal to the Bayes conditional probability given the corresponding information set. Implications are as follows. First, each posterior is uniquely determined. Second, qualitative belief reduces to fully introspective knowledge in a "standard" environment. Thus, a care must be taken when one studies non-veridical belief or non-introspective knowledge. Third, an information partition compatible with the consistency conditions is uniquely determined by the posteriors. Fourth, qualitative and probability-one beliefs satisfy truth axiom almost surely. The paper also sheds light on how the additivity of the posteriors yields negative introspective properties of beliefs. * I would like to thank Nobuo Koida, Massimo Marinacci, and Sujoy Mukerji for their insightful comments and discussions. I would also like to thank the TARK reviewers for their insightful comments.1 See, for instance, [11] for the importance of capturing both knowledge and probabilistic beliefs. See, for example, [49] for using qualitative and probabilistic beliefs for studying solution concepts of games.On the Consistency among Prior, Posteriors, and Information Sets of each posterior affects her reasoning, for now I assume each posterior to be countably additive. On the other hand, Alameda has a mapping, called a possibility correspondence. It associates, with each state, the set of states that she considers possible (the information set) at that state. I, as an analyst, derive properties of information sets, instead of directly assuming them. The framework is fairly parsimonious.The main result (Theorem 1) restates the consistency conditions as: (i) Alameda's information sets form a partition almost surely; and (ii) her posterior at each state coincides with the Bayes conditional probability given her corresponding information set. While information sets are usually exogenously assumed to form a partition, the main result demonstrates that the consistency conditions on the agent's qualitative and quantitative beliefs alone determine Bayes updating.While the main result has its own interest, I rather use it to derive the following four implications. The first implication (Corollary 1) is that the consistency conditions uniquely determine the posterior at each state as the Bayes conditional probability given the associated information set.The second implication (Corollary 2) is on the introspective properties of qualitative belief. Alameda's introspective abilities in her qualitative belief are reflected in properties of her information sets. When her information sets form a partition, her qualitative belief becomes knowledge (true belief) with full introspection. Truth axiom obtains: she ca...