Objective Bayesian calibration and the problem of non-convex evidence

Jon Williamson’s Objective Bayesian Epistemology relies upon a calibration norm to constrain credal probability by both quantitative and qualitative evidence. One role of the calibration norm is to ensure that evidence works to constrain a convex set of probability functions. This essay brings into focus a problem for Williamson’s theory when qualitative evidence specifies non-convex constraints. Jon Williamson (2010) provides a spirited defense of a version of Objective Bayesianism which relies upon a calibration norm to constrain credal probability by evidence. According to this norm, an agent’s degrees of belief should be constrained by two types of evidence: quantitative evidence, which directly constrains admissible values of chance functions, and qualitative evidence, such as logical or causal constraints on chance variables, which may indirectly constrain chance functions. Once those constraints are in place, the theory maintains that the agent’s degrees of belief should be maximally equivocal between the basic outcomes. I have discussed the calibration norm with sympathy (Wheeler and Williamson 2011), although in that setting our aim was to reconcile the reference class reasoning machinery of Evidential Probability (Kyburg and Teng 2001) with Williamson’s Objective Bayesian Epistemology (OBE), so we naturally played down the differences between the two theories. Nevertheless, I have reservations about viewing OBE as a general theory for rational belief. So, in response to Williamson’s call to go whole hog, I would like to bring into focus a problem for OBE that arises when qualitative evidence prescribes non-convex constraints on a set of chance functions. The mechanics of OBE depend on a convex set of probability functions, and one role the calibration norm plays is to enforce this convexity condition. This can be a reasonable approach when the endpoints of an interval designate upper and lower bounds on admissible degrees of belief and the question to answer is what point within this range represents the most cautious position for an agent to take. That is the question that OBE is set up to answer, and it does so by advising the agent to pick the most equivocal point within the convex hull of admissible options D ra ft of Ju ly 12 ,2 01 1 delineated by the calibration norm. In practice this means that OBE shuns both high and low probabilities within a given set when a more equivocal alternative is available. The problem is that not all evidence fits the OBE mold. In particular, non-convex evidence can reverse the order of what is “extreme” and what is “cautious” to believe, throwing the weight of the theory behind precisely the wrong candidates for rational belief. What is more, the calibration postulates only give a definition of calibrated sets of probability functions rather than an algorithm for how to construct those sets. As it turns out, applying the calibration norm is a case-by-case exercise, one that depends crucially on judgments about how to model quantitative and qualitative evidential constraints. The instrumental role that interpretation plays, absent a general method for calibrating evidence, threatens to render OBE objective in name only. As a preview of what is to come, we first recount Williamson’s postulates for calibration and then demonstrate that the postulates cannot be interpreted as a procedure for constructing calibrated sets of probability functions. Then we demonstrate that the calibration norm cannot be interpreted as a general norm for characterizing evidence by showing that calibration fails to correctly constrain degrees of belief when evidence is non-convex. Finally, in closing, we observe that despite claims to the contrary, OBE fails to clearly distinguish between known evidence and belief. 1 Formal Preliminaries The strength of an agent’s belief is assumed to be representable by a probability function, P, which is defined in standard form with respect to a probability space (Ω,F ,P) such that F is a σ -algebra over a set Ω and P : F −→ [0,1] is a probability measure defined on the space (Ω,F ,P) satisfying