Imprecise Hypothesis-Based Bayesian Decision Making with Simple Hypotheses
Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications, PMLR 103:338-345, 2019.
Applied real-world decisions are frequently guided by the outcome of hypothesis-based statistical analyses. However, most often relevant information about the phenomenon of interest is available only imprecisely, and misleading results might be obtained, in particular, by either ignoring relevant information or pretending a level of knowledge that is not given. In order to be able to include (partial) information authentically in the imprecise form it is available, this paper tries to extend the framework of hypothesis-based Bayesian decision making with simple hypotheses to be able to deal with imprecise information about the three relevant quantities: hypotheses, prior beliefs, and loss function. Although straightforward at first glance, it appears that by specifying the hypotheses imprecisely, Bayesian updating of the prior beliefs might be inconsistent. In that, this paper provides the basic mathematical formulation to further extend imprecise hypothesis-based Bayesian decision theory to more elaborate contexts, such as those involving composite imprecise hypotheses, and in addition highlights the necessity of paying particular attention to the depicted updating issues.