The Maximum Entropy of the Mean (MEM) is a general approach to statistical estimation dating back to ideas of Jaynes in 1957. We begin by exploring MEM as a regularization method in the blind deblurring and denoising of images containing some form of symbology; for example, QR bar codes. We then address the MEM in far more generality, focusing on the relation between the MEM function (which is defined via an infinite-dimensional variational problem) and the Cramer rate function famous in large deviation theory. In doing so, we see how, for a variety of known prior distributions, the MEM function can be explicitly computed and efficiently implemented in a variety of models. Finally, we return to the MEM as a method for image denoising to show how it lends itself to coupling with a data-driven prior, computed via a deep neural network. Collaborators in various parts of this talk are: Former McGill undergraduates Gabriel Rioux (currently a PhD student at Cornell) and Christopher Scarvelis (currently a PhD student at MIT); Tim Hoheisel (McGill), Yakov Vaisbourd (McGill), current McGill undergraduate Ariel Goodwin, Pierre Marechal (Toulouse), and Carola-Bibiane Schonlieb (Cambridge); Current McGill undergraduate Jack-Richter Powell.