The α-divergences associated with a pair of strictly comparable quasi-arithmetic means

Abstract

We generalize the family of $\alpha$-divergences using a pair of strictly comparable weighted means. In particular, we obtain the $1$-divergence in the limit case $\alpha \to 1$ (a generalization of the Kullback-Leibler divergence) and the $0$-divergence in the limit case $\alpha \to 0$ (a generalization of the reverse Kullback-Leibler divergence). We state the condition for a pair of quasi-arithmetic means to be strictly comparable, and report the formula for the quasi-arithmetic $\alpha$-divergences and its subfamily of bipower homogeneous $\alpha$-divergences which belong to the Csisár's $f$-divergences. Finally, we show that these generalized quasi-arithmetic $1$-divergences and $0$-divergences can be decomposed as the sum of generalized cross-entropies minus entropies, and rewritten as conformal Bregman divergences using monotone embeddings.

18 pages

References in corpus (1)

• Bahadur efficiency of the maximum likelihood estimator and one-step estimator for quasi-arithmetic means of the Cauchy distribution