Relative $alpha$-Entropy Minimizers Subject to Linear Statistical Constraints

We study minimization of a parametric family of relative entropies, termed relative $alpha$-entropies (denoted $mathscr{I}_{alpha}(P,Q)$). These arise as redundancies under mismatched compression when cumulants of compressed lengths are considered instead of expected compressed lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative $alpha$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimization of $mathscr{I}_{alpha}(P,Q)$ over the first argument on a set of probability distributions that constitutes a linear family is studied. Such a minimization generalizes the maximum R{e}nyi or Tsallis entropy principle. The minimizing probability distribution (termed $mathscr{I}_{alpha}$-projection) for a linear family is shown to have a power-law.