Minimaximax approach for finding optimal decisions’ subset regarding changes of the loss function

Abstract

A generalization of the decision (loss or utility) function is suggested. An ordinary decision function is defined on a Cartesian product of a decisions’ set and a set of states, but the generalized decision function has the third variable called a metastate. Metastates are generated due to uncertain evaluation of ordinary situations, or influence of the time course. For minimizing losses under poor or unreliable statistics, the rule of minimaximax is fully described. For correctly transferring from minimaximax to Bayesian criterions, the rules of minimizing expected losses for the generalized loss function are formalized. All the suggested criterions are re-formalized for the case of the utility function.