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

## TL;DR

This paper explores the implications of consistency conditions among priors, posteriors, and information sets, revealing their structural properties and impact on belief and knowledge introspection in probabilistic frameworks.

## Contribution

It reformulates consistency conditions as partitioning of information sets and equality of posteriors to Bayes' rule, clarifying belief and knowledge properties.

## Key findings

- Posteriors are uniquely determined by information sets.
- Qualitative belief reduces to fully introspective knowledge under certain conditions.
- Consistency conditions determine a unique information partition.

## 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.