Probabilistic Belief Contraction: Considerations on Epistemic Entrenchment, Probability Mixtures and KL Divergence

Abstract

Probabilistic belief contraction is an operation that takes a probability distribution P representing a belief state along with an input sentence a representing some information to be removed from this belief state, and outputs a new probability distribution $P^-_a$. The contracted belief state $P^-_a$ can be represented as a mixture of two states: the original belief state P, and the resultant state $P^*_{\lnot a}$ of revising P by $\lnot a$. Crucial to this mixture is the mixing factor $\epsilon$ which determines the proportion of P and $P^*_{\lnot a}$ that are used in this process in a uniform manner. Ideas from information theory such as the principle of minimum cross-entropy have previously been used to motivate the choice of the probabilistic contraction operation. Central to this principle is the Kullback-Leibler (KL) divergence. In an earlier work we had shown that the KL divergence of $P^-_a$ from P is fully determined by a function whose only argument is the mixing factor $\epsilon$. In this paper we provide a way of interpreting $\epsilon$ in terms of a belief ranking mechanism such as epistemic entrenchment that is in consonance with this result. We also provide a much needed justification for why the mixing factor $\epsilon$ must be used in a uniform fashion by showing that the minimal divergence of $P^-_{a}$ from P is achieved only when uniformity is respected.