From Optimal Inductive Methods to Optimal Beliefs

Abstract

Abstract Chapter 7 begins with the application of meta-induction to the prediction of probability distributions, in the form of probabilistic prediction games. The attractivity-weighted aggregation of candidate distributions yields a predictively optimal probability distribution. The adoption of this distribution as one’s rational degree of belief is justified by the optimality principle, since acting according to this distribution maximizes expected utility. By tracking the success of distributions over games with varying event sequences a meta-inductive a posteriori justification of inductive generalizations in the form of exchangeability assumptions is possible. Section 7.3 is devoted to the optimality of the principle of total evidence as an important complement to methods of induction. It is shown that conditionalizing degrees of belief on one’s total relevant evidence may only increase, not decrease, one’s expected success. The two final sections deal with the justificational transition from rational degrees of beliefs to qualitative (yes-or-no) beliefs. The question of when it is rational to accept a highly probable proposition as a qualitative belief leads into difficult problems involving a clash between different rationality principles, such as Locke’s condition and the principle of conjunctive closure. This clash is exemplified in the lottery paradox and the paradox of the preface. It is argued that the optimal strategy for extracting qualitative beliefs out of rational degrees of belief depends on the context: while for success-essential beliefs a high probability of strict truth is mandatory, for global theories a high probability of approximate truth is sufficient.