Minimization of divergences on sets of signed measures

1 LSTA - Laboratoire de Statistique Théorique et Appliquée 2 LMR - Laboratoire de Mathématiques de Reims Reims

Abstract : We consider the minimization problem of $\phi$-divergences between a given probability measure $P$ and subsets $\Omega$ of the vector space $\mathcal{M} \mathcal{F}$ of all signed finite measures which integrate a given class $\mathcal{F}$ of bounded or unbounded measurable functions. The vector space $\mathcal{M} \mathcal{F}$ is endowed with the weak topology induced by the class $\mathcal{F}\cup \mathcal{B} b$ where $\mathcal{B} b$ is the class of all bounded measurable functions. We treat the problems of existence and characterization of the $\phi$-projections of $P$ on $\Omega$. We consider also the dual equality and the dual attainment problems when $\Omega$ is defined by linear constraints.

Keywords : Empirical Likelihood Moment Problem Convex Programming Maximum Entropy Fenchel Duality Minimum Divergences Convex Distances Fenchel Duality.

Author: Michel Broniatowski - Amor Keziou -

Source: https://hal.archives-ouvertes.fr/