Probabilistic Stability: dynamics, nonmonotonic logics, and stable revision
Krzysztof Mierzewski
Abstract:
Leitgeb proposes an acceptance rule based on the notion of probabilistically stable hypotheses. This stability rule offers a formal solution to the Lottery Paradox and suggests a promising account of the relationship between logical and probabilistic models of belief. In this thesis, we investigate the role of probabilistic stability in bridging logical information dynamics – modeled by revision operators – with probabilistic models of belief change, as captured by Bayesian conditioning.
Our first topic is the connection between Bayesian conditioning and AGM revision operators. The gold standard of dynamic compatibility between a logical revision operator and Bayesian conditioning is given by the tracking criterion, which amounts to the requirement that the revision operator commmute with Bayesian update modulo the acceptance rule.
A general impossibility theorem by Lin & Kelly shows that no well-behaved acceptance rule allows AGM operators to track Bayesian update. We show that, even though Leitgeb’s stability rule falls prey to Lin & Kelly’s theorem, there is nonetheless a precise sense in which it allows to bridge AGM revision and conditioning. We establish this by appealing to notions from information theory: by an application of the principle of maximum entropy, we show that AGM revision operators can be generated, through Leitgeb’s rule, by Bayesian conditioning. In situations of information loss, AGM revision is compatible with – and indeed emerges from – Bayesian conditioning.
Another approach to the tracking problem is to axiomatise the revision operators generated by the stability rule: the study of these probabilistically stable revision operators constitutes our second topic. We show that the class of probabilistically stable revision operators can be captured using selection function models, as employed in non-monotonic logics. We first identify the key properties of the resulting non-monotonic logic. We then prove a probabilistic representation theorem for the selection function models in question. The theorem, which draws on the theory of comparative probability orders, yields a complete characterisation of probabilistically stable revision operators. Along the way, we prove a general result giving sufficient conditions for the joint representation of a pair of (respectively, strict and non-strict) comparative probability orders, and we point out an application of the representation theorem to simple voting games.