Properties and Inequalities for φ-entropies Derived from Inverse MaxEnt Problems

International audienceThis paper focuses on $\phi$-entropy functionals derived from a MaxEnt inverse problem. This consists in determining an entropy, given a target maximum entropy distribution and given (usually simple) moment constraints. This kind of problem, considered, e.g., by Kesavan (1989), complements the usual maxent inverse problem where one looks for the constraints that will lead to a target maxent distribution, given the entropy. This approach allows distributions to be considered outside the exponential family—to which the maximizers of the Shannon entropy belong, and also to consider simple moment constraints, which can be estimated from the observed sample.We generalize Kesavan's approach and express the solutions and their properties in a broad setting. Our approach also yields entropic functionals that are functions of both probability density and state, allowing us to include skew-symmetric or multimodal distributions in the setting.In the classical setting, there is an interplay between information measures (entropy, Fisher information) and moments. Associated to the generalized $\phi$-entropies with a prescribed maxen distribution, we introduce informational quantities such as a generalized escort distribution, generalized moments and generalized Fisher information. These generalized informationalquantities then allow the usual informational relations to be extended such as the Cram\'er-Rao inequality and the de Bruijn identity, relations saturated (or valid) precisely for the generalized MaxEnt distributions. Of course, classical results for Shannon and Rényi–Tsallis entropies are included as special cases