On relative entropy projection algorithms with marginal constraints.

Summary This thesis is centered around an algorithm for the construction of probability measures with prescribed marginal laws, called Iterative Proportional Fitting (IPF). Coming from statistics, this algorithm is based on successive projections on probability spaces with the Kullback-Leibler pseudo relative entropy distance. This thesis constitutes an overview of the available results on the subject, and contains some extensions and refinements. The first part is devoted to the study of relative entropy projections, to existence and uniqueness criteria as well as to characterization criteria related to the closure of a sum of subspaces. Under certain conditions, the problem becomes a maximum entropy problem for graphical marginal constraints. The second part highlights the iterative IPF process. Originally addressing an estimation problem for contingency tables, it is more generally an analogue of a classical algorithm of alternating projections on Hilbert spaces. After presenting the properties of the IPF, we focus on convergence results in the discrete finite case and in the Gaussian case, as well as in the continuous case with two marginals, for which an extension is proposed. We then focus on the Gaussian case, for which a new formulation of the IPF allows us to obtain a speed of convergence in the case with two prescribed marginals, whose optimality in dimension 2 is shown.