Data-based decisions under imprecise probability and least favorable models

Data-based decision theory under imprecise probability has to deal with optimization problems where direct solutions are often computationally intractable. Using the Γ-minimax optimality criterion, the computational effort may significantly be reduced in the presence of a least favorable model. Buja [A. Buja, Simultaneously least favorable experiments. I. Upper standard functionals and sufficiency, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 65 (1984) 367–384] derived a necessary and sufficient condition for the existence of a least favorable model in a special case. The present article proves that essentially the same result is valid in case of general coherent upper previsions. This is done mainly by topological arguments in combination with some of Le Cam’s decision theoretic concepts. It is shown how least favorable models could be used to deal with situations where the distribution of the data as well as the prior is allowed to be imprecise.