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تسجيل مستخدم جديدRelative $alpha$-Entropy Minimizers Subject to Linear Statistical Constraints
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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.
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Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative $alpha$-entropies (denoted $mathscr{I}_{alpha}$), arise as redundancies under mismatched comp
ression 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. Minimizers of these relative $alpha$-entropies on closed and convex sets are shown to exist. Such minimizations generalize the maximum R{e}nyi or Tsallis entropy principle. The minimizing probability distribution (termed forward $mathscr{I}_{alpha}$-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 $mathscr{I}_{alpha}$-projection is studied.
In part I of this two-part work, certain minimization problems based on a parametric family of relative entropies (denoted $mathscr{I}_{alpha}$) were studied. Such minimizers were called forward $mathscr{I}_{alpha}$-projections. Here, a complementary
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We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these
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