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

We study minimization of a parametric family of relative entropies, termed relative $α$-entropies (denoted $\mathscr{I}_α(P,Q)$). 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 $α$-entropies behave like squared Euclidean distance and satisfy the Pythagorean property. Minimization of $\mathscr{I}_α(P,Q)$ over the first argument on a set of probability distributions that constitutes a linear family is studied. Such a minimization generalizes the maximum Rényi or Tsallis entropy principle. The minimizing probability distribution (termed $\mathscr{I}_α$-projection) for a linear family is shown to have a power-law.