Invariant Equivocation

Abstract

Objective Bayesians hold that degrees of belief ought to be chosen in the set of probability functions calibrated with one's evidence. The particular choice of degrees of belief is via some objective, i.e., not agentdependent, inference process that, in general, selects the most equivocal probabilities from among those compatible with one's evidence. Maximising entropy is what drives these inference processes in recent works by Williamson and Masterton though they disagree as to what should have its entropy maximised. With regard to the probability function one should adopt as one's belief function, Williamson advocates selecting the probability function with greatest entropy compatible with one's evidence while Masterton advocates selecting the expected probability function relative to the density function with greatest entropy compatible with one's evidence. In this paper we discuss the significant relative strengths of these two positions. In particular, Masterton's original proposal is further developed and investigated to reveal its significant properties; including its equivalence to the centre of mass inference process and its ability to accommodate higher order evidence.Keywords: Maximum Entropy, Objective Bayesianism, Inference Process, Centre of Mass, Language Invariance 1 Introduction "How should one form graded beliefs?" is a question that has long fascinated philosophers. The answer to this question is highly relevant throughout science, law, operational research and policy-making. Intuitively, it is obvious that one's evidence ought to matter when forming beliefs, whether graded or binary. The best way of caching out this ubiquitous intuition is, however, a matter of lively disagreement. The main philosophical protagonists in this debate are: the subjective Bayesians, who can be further subdivided into the radical (de Finetti This paper is concerned with objective Bayesianism. Objective Bayesians, like their subjective brethren, hold that one's degrees of belief ought to be consistent with one's evidence. However, in cases where the evidential constraints are satisfied by more than one probability function the objective Bayesians differ from the subjective Bayesians in holding that the choice of a particular probability function from those consistent with the evidential constraints ought to made 1 in an objective-i.e., agent-independent-manner. Typically, such a choice is made by applying an inference process that picks from among the probability functions consistent with the evidential constraints that function which is, in some sense, maximally equivocal. 1 Objective Bayesianism is (still!) a minority view with a relatively small but dedicated group of advocates. Even in this relatively small group there are disagreements. One such recent disagreement is that between Williamson [27] and Masterton [16] on how to understand maximal equivocation.We shall herein first rehearse Williamson's account in Section 2.1 and 2.2. Then we improve on Masterton's approach by reformulating it, Section 2...