For Bayesians, Rational Modesty Requires Imprecision

Abstract

G Belot (2013) has recently developed a novel argument against Bayesianism. He shows that there is an interesting class of problems that, intuitively, no rational belief forming method is likely to get right. But a Bayesian agent’s credence, before the problem starts, that she will get the problem right has to be 1. This is an implausible kind of immodesty on the part of Bayesians.1 My aim is to show that while this is a good argument against traditional, precise Bayesians, the argument doesn’t neatly extend to imprecise Bayesians. As such, Belot’s argument is a reason to prefer imprecise Bayesianism to precise Bayesianism. For present purposes, the precise Bayesian agent has just two defining characteristics. First, their credences in all propositions are given by a particular countably additive probability function. Second, those credences are updated by conditionalisation as new information comes in. These commitments are quite strong in some respects. They say that there is a single probability function that supplies the agent’s credences no matter which question is being investigated, and no matter how little evidence the agent has before the investigation is started. The everyday statistician, even one who is sympathetic to Bayesian approaches, may feel no need to sign up for anything this strong. But many philosophers seem to be interested in varieties of Bayesianism that are just this strong. For instance, there has been extensive discussion in recent epistemology of whether various epistemological approaches, such as dogmatism, can be