2014/10/09 by M. Ashok Kumar, Kumar, M. Ashok, Rajesh Sundaresan +1
Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Statistical Mechanics and Entropy #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1410.2346
openalex publication_date 2014/10/09 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative α-entropies (denoted \mathscrIα), 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. Minimizers of these relative α-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum Rényi or Tsallis entropy principle. The minimizing probability distribution (termed forward \mathscrIα-projection) for a linear family is shown to obey a power-law. Other results in connection with statistical inference, namely subspace transitivity and iterated projections, are also established. In a companion paper, a related minimization problem of interest in robust statistics that leads to a reverse \mathscrIα-projection is studied.