The Modal Logic of Bayesian Belief Revision

Abstract

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.

Standard Bayes Logic Is Not Finitely Axiomatizable

“…We start by recalling definitions and theorems from [11] and [2]. Medvedev's logic of finite problems and its extension to infinite problems by Skvortsov originate in intuitionistic logic.…”

“…. A family of normal modal logics have been defined in [2] based on finite or countable or countably infinite or all Bayes frames as follows.…”

“…As a slight abuse of notation we will use the term Medvedev frame for any frame of the form P 0 (X) (thus X need not be finite here). One of the main results in [2] is Theorem 5.2:…”

“…provided p(A) = 0. The paper [2] aimed at studying the logical aspects of this type of inference from the perspective of modal logic. 1 Note that if the measurable space X, B is finite or countable, then B is the full powerset algebra ℘(X) (we rely on the convention that elementary events {x} for x ∈ X always belong to the algebra B).…”

Having a look at the Bayes Blind Spot

“…This is in harmony with the fact that Bayesian learning understood as statistical inference via conditionalization is an ampliative inference, not a deductive one. This feature Bayesian inference also is reflected by the non-axiomatizability of certain modal logics that are defined semantically in terms of the Bayes accessibility relation (Brown et al 2018).…”