### A Philosopher's View on Bayesian Evaluation of Informative Hypotheses [chapter]

Jan-Willem Romeijn, Rens van de Schoot
2008 Bayesian Evaluation of Informative Hypotheses
1 Philosophical Analysis of Bayesian Model Selection 3 of reasoning: deductive logic. In Section 1.4, based on this parallel, we will describe the role of a statistical model in a Bayesian statistical inference as a specific type of premise in an inductive. We can thereby identify elements of the views of both Popper and Carnap in Bayesian statistical inference and extend Bayesian inference to model selection, in particular, the selection by means of Bayes factors. This leads to a discussion of
more » ... to a discussion of some problematic aspects of Bayesian model selection procedures in Section 1.5. We will address two specific worries. First, a comparison of models in terms of their posterior model probabilities does not seem to make sense if the models overlap. We will remedy this by organizing the space on which the models are defined a bit differently. Second, and in view of this reorganization, we ask how we can interpret the probability assignments to hypotheses. Statistics and the Problem of Induction This section deals with statistics, its relation to the problem of induction, and the solutions that Popper and Carnap provided for this problem, drawing on standard textbooks in the philosophy of science such as Bird  and Curd and Cover . We will see that these solutions, in this context termed inductivism and rationalism, are endpoints in a spectrum of positions and that, as such, they both miss out on an important aspect of statistical reasoning. The Problem of Induction Induction is a mode of inference that allows us to move from observed data to as yet unknown data elements and empirical generalizations. A typical example of an inductive inference is presented in Statements 1 and 2: