Symmetry and partial belief geometry

Abstract

When beliefs are quantified as credences, they are related to each other in terms of closeness and accuracy. The "accuracy first" approach in formal epistemology wants to establish a normative account for credences (probabilism, Bayesian conditioning, principle of indifference, and so on) based entirely on the alethic properties of the credence: how close it is to the truth. To pull off this project, there is a need for a scoring rule. There is widespread agreement about some constraints on this scoring rule (for example propriety), but not whether a unique scoring rule stands above the rest. The Brier score equips credences with a structure similar to metric space and induces a "geometry of reason." It enjoys great popularity in the current debate. I point out many of its weaknesses in this article. An alternative approach is to reject the geometry of reason and accept information theory in its stead. Information theory comes fully equipped with an axiomatic approach which covers probabilism, standard conditioning, and Jeffrey conditioning. It is not based on an underlying topology of a metric space, but uses a non-commutative divergence instead of a symmetric distance measure. I show that information theory, despite initial promise, also fails to accommodate basic epistemic intuitions; and speculate on its remediation.