Minimax Entropy and Maximum Likelihood. Complementarity of tasks, identity of solutions

Abstract

Concept of exponential family is generalized by simple and general exponential form. Simple and general potential are introduced. Maximum Entropy and Maximum Likelihood tasks are defined. ML task on the simple exponential form and ME task on the simple potentials are proved to be complementary in set-up and identical in solutions. ML task on the general exponential form and ME task on the general potentials are weakly complementary, leading to the same necessary conditions. A hypothesis about complementarity of ML and MiniMax Entropy tasks and identity of their solutions, brought up by a special case analytical as well as several numerical investigations, is suggested in this case. MiniMax Ent can be viewed as a generalization of MaxEnt for parametric linear inverse problems, and its complementarity with ML as yet another argument in favor of Shannon's entropy criterion.