  We explore a method of statistical estimation called Maximum Entropy on the
Mean (MEM) which is based on an information-driven criterion that quantifies
the compliance of a given point with a reference prior probability measure. At
the core of this approach lies the MEM function which is a partial minimization
of the Kullback-Leibler divergence over a linear constraint. In many cases, it
is known that this function admits a simpler representation (known as the
Cram\'er rate function). Via the connection to exponential families of
probability distributions, we study general conditions under which this
representation holds. We then address how the associated MEM estimator gives
rise to a wide class of MEM-based regularized linear models for solving inverse
problems. Finally, we propose an algorithmic framework to solve these problems
efficiently based on the Bregman proximal gradient method, alongside proximal
operators for commonly used reference distributions. The article is
complemented by a software package for experimentation and exploration of the
MEM approach in applications.
