In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes' rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent's prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal companion of Medvedev's logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable.

No abstract

A classical probability measure space was defined in earlier papers [14], [9] to be common cause closed if it contains a Reichenbachian common cause of every correlation in it, and common cause incomplete otherwise. It is shown that a classical probability measure space is common cause incomplete if and only if it contains more than one atom. Furthermore, it is shown that every probability space can be embedded into a common cause closed one; which entails that every classical probability space is common cause completable with respect to any set of correlated events. The implications of these results for Reichenbach's Common Cause Principle are discussed, and it is argued that the Principle is only falsifiable if conditions on the common cause are imposed that go beyond the requirements formulated by Reichenbach in the definition of common cause.