On the Equivalence between Set--Theoretic and Maxent MAP Estimation

by Prakash Ishwar and Pierre Moulin

In this paper, we establish an equivalence between two conceptually
different methods of signal estimation under modeling uncertainty viz.
set--theoretic estimation and maximum entropy (maxent) MAP estimation.
The first method assumes constraints on the signal to be estimated,
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 set--theoretic 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.