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

• Main Authors: Kumar, M. Ashok, Sundaresan, Rajesh
• Format: Journal Article
• Language: English
• Published: 18.10.2014
• Subjects: Computer Science - Information Theory; Mathematics - Information Theory; Mathematics - Statistics Theory; Statistics - Theory
• DOI: 10.48550/arxiv.1410.4931

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 Rényi or Tsallis entropy principle. The minimizing probability distribution (termed $\mathscr{I}_{\alpha}$-projection) for a linear family is shown to have a power-law.