Imprecise Hypothesis-Based Bayesian Decision Making with Composite Hypotheses

Statistical analyses with composite hypotheses are omnipresent in empirical sciences, and a decision-theoretic account is required in order to formally consider their practical relevance. A Bayesian hypothesis-based decision-theoretic analysis requires the specification of a prior distribution, the hypotheses, and a loss function, and determines the optimal decision by minimizing the expected posterior loss of each hypothesis. However, specifying such a decision problem unambiguously is rather difficult as, typically, the relevant information is available only partially. In order to include such incomplete information into the analysis and to facilitate the use of decision-theoretic approaches in applied sciences, this paper extends the framework of hypothesis-based Bayesian decision making with composite hypotheses into the framework of imprecise probabilities, such that imprecise specifications for the prior distribution, for the composite hypotheses, and for the loss function are allowed. Imprecisely specified composite hypotheses are sets of parameter sets that are able to incorporate blurring borders between hypotheses into the analysis. The imprecisely specified prior distribution gets updated via generalized Bayes rule, such that imprecise probabilities of the (imprecise) hypotheses can be calculated. These lead – together with the (imprecise) loss function – to a set-valued expected posterior loss for finding the optimal decision. Beneficially, the result will also indicate whether or not the available information is sufficient to guide the decision unambiguously, without pretending a level of precision that is not available.