On the equivalence of set-theoretic and maxent MAP estimation

In this paper, we establish an equivalence, between two conceptually different methods of signal estimation under modeling uncertainty viz. set-theoretic (ST) estimation and maximum entropy (maxent) MAP estimation. The first method assumes constraints on the signal to be estimated, and the second assumes constraints on a probability distribution for the signal. We provide broad conditions under which the two aforementioned estimation paradigms produce the same signal estimate. We also show how the maxent formalism can be used to provide solutions to three important problems: How to select sizes of constraint sets in ST estimation (the analysis highlights the role of shrinkage); how to choose the values of parameters in regularized restoration when using multiple regularization functionals; and how to trade off model complexity and goodness of fit in a model selection problem.