Coherent States and Bayesian Duality

We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions. Department of Mathematics and Statistics, Concordia University, Montréal, Québec, CANADA H3G 1M8 e-mail: [email protected] Astroparticules et Cosmologie (APC, UMR 7164), Université Paris Diderot Paris 7, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13 e-mail: [email protected] 3 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616 e-mail: [email protected], [email protected]