Belief, Evidence, and Conditioning

Abstract

Since Ramsey, much discussion of the relation between probability and belief has taken for granted that there are degrees of belief, i.e., that there is a real-valued function, B, that characterizes the degree of belief that an agent has in each statement of his language. It is then supposed that B is a probability. It is then often supposed that as the agent accumulates evidence, this function should be updated by conditioning:$B^{E}(\cdot) $ should be$B(\cdot \wedge E) / B(E) $ . Probability is also important in classical statistics, where it is generally supposed that probabilities are frequencies, and that inference proceeds by controlling error and not by conditioning. I will focus on the tension between these two approaches to probability, and in the main part of the paper show where and when Bayesian conditioning conflicts with error based statistics and how to resolve these conflicts.

Does the Dome Defeat the Material Theory of Induction?

“…Norton's use of direct inference in his bismuth example, he has not yet explicitly discussed this worry. Here is a response that is similar to Henry E. Kyburg's treatment of sampling distributions in his theory of direct inference (Kyburg, 2006). It is true that knowledge of more informative sampling distributions can undermine the inferences recommended by direct inference from more general combinatoric principles.…”

Probability and Statistics in the Tinbergen-Keynes Debates

“…It is hence unsurprising that philosophers and statisticians have been interested in combining these methodologies. For instance, some contemporary philosophers (Kyburg and Teng 2001;Kyburg 2006;Williamson 2010Williamson , 2013 have proposed an intriguing compromise, which I shall call Objective Bayesianism. 13 According to this approach, Bayesian reasoning takes precedence over frequentist reasoning, but only when the 'priors' involved are derivable from background knowledge about the phenomena.…”

Evidentialism, Inertia, and Imprecise Probability

“…Kyburg uses the term "Sharpening" for this process of ignoring information in our premises. The details of Sharpening evolved over time; I shall use his last version (Kyburg, 2006). To have a Kyburgian probability, our hypothesis of interest must be equivalent to a class-membership statement for a single-case.…”