Simplicity in Bayesian nested-model comparisons: Popper’s disagreement with Wrinch and Jeffreys revisited

Bayesian nested-model comparisons involve an assessment of the probabilities for a relatively simple model and a more general encompassing model. Since the simpler model can be viewed as a subset of the more complex model it is nested in, Popper has argued that the axioms of probability are violated when the simpler model is nonetheless assigned a higher prior probability. While Popper raised this objection in the context of assigning prior probabilities to models, we argue that Popper’s objection does not just concern the priors, but Bayesian model comparisons more generally. We term this ‘the subset problem’. A variety of solutions have been proposed in the literature. We discuss some of these solutions and combine them into a new Bayesian account, in which both the probability assignments and the algebra over which they are assigned receive a specific interpretation. Finally, we discuss a new non-Bayesian solution, in which nested models are assigned an attractiveness measure that need not be additive.