In this paper, we study linear inverse problems where some generalized moments of an unknown positive measure are observed. We introduce a new construction, called the maximum entropy on the mean method (MEM), which relies on a suitable sequence of finite-dimensional discretized inverse problems. Its advantage is threefold: It allows us to interpret all usual deterministic methods as Bayesian methods; it gives a very convenient way of taking into account prior information; it also leads to new criteria for the existence question concerning the linear inverse problem which will be a starting point for the investigation of superresolution phenomena. The key tool in this work is the large deviations property of some discrete random measure connected with the reconstruction procedure.
"Bayesian methods and maximum entropy for ill-posed inverse problems." Ann. Statist. 25 (1) 328 - 350, February 1997. https://doi.org/10.1214/aos/1034276632