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.