Examine a method of statistical estimation called maximum entropy of the mean (MEM). It 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 is the MEM function, which is a partial minimization of the Kullback-Leibler divergence over linear constraints. In many cases, this function is known to allow simpler representations (known as the Cram\’er rate function). We study the general conditions under which this representation holds through its connection to the exponential family of probability distributions. We then describe how the associated MEM estimator produces a wide class of His MEM-based regularized linear models for solving inverse problems. Finally, we propose an algorithmic framework for efficiently solving these problems based on the Bregman proximal gradient method, together with commonly used proximal operators of reference distributions. This article is complemented by a software package for experimentation and investigation of his MEM approach in applications.