Bayesian learning for a class of priors with prescribed marginals

Abstract

We discuss three learning rules for generalized Bayesian updating of an imprecise probability: (a modified version of) the generalized Bayes’ rule, the maximum likelihood update rule (after Gilboa and Schmeidler) and a newly developed hybrid rule. We investigate the general methodology for a special class of multivariate probability measures with prescribed marginals but arbitrary correlation structure. Both the choice and analysis of this class are motivated by expert interviews that we conducted with modelers in the field of climatic change. We argue that both updating rules from the literature have strong limitations, the generalized Bayes’ rule is too conservative, i.e., too inclusive, while the maximum likelihood update rule being too exclusive, adding spurious information. As a powerful extension we introduce a new rule for Bayesian updating of an imprecise measure: a “weighted likelihood update method,” which bases Bayesian updating on the whole set of priors but weights the influence of its members. We study the different rules in the case of bivariate Gaussian priors. Our investigation shows that the new rule combines certain attractive features of the generalized Bayes’ rule and the maximum likelihood update rule. In this article we aim at highlighting the sequence of not yet fully resolved statistical issues a practitioner on complex mechanistic models would face when updating imprecise prior knowledge.