Relative α-entropy minimizers subject to linear statistical constraints

We study minimization of a parametric family of relative entropies, termed relative α-entropies (denoted 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 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 ℐ_α-projection) for a linear family is shown to have a power-law.