On Divergences and Informations in Statistics and Information Theory

The paper deals with the
$f$
-divergences of CsiszÁr generalizing the discrimination information of Kullback, the total variation distance, the Hellinger divergence, and the Pearson divergence. All basic properties of
$f$
-divergences including relations to the decision errors are proved in a new manner replacing the classical Jensen inequality by a new generalized Taylor expansion of convex functions. Some new properties are proved too, e.g., relations to the statistical sufficiency and deficiency. The generalized Taylor expansion also shows very easily that all
$f$
-divergences are average statistical informations (differences between prior and posterior Bayes errors) mutually differing only in the weights imposed on various prior distributions. The statistical information introduced by De Groot and the classical information of Shannon are shown to be extremal cases corresponding to
$alpha =0$
and
$alpha =1$
in the class of the so-called Arimoto
$alpha $
-informations introduced in this paper for
$0≪ alpha ≪ 1$
by means of the Arimoto
$alpha $
-entropies. Some new examples of
$f$
-divergences are introduced as well, namely, the Shannon divergences and the Arimoto
$alpha $
-divergences leading for
$alpha uparrow 1$
to the Shannon divergences. Square roots of all these divergences are shown to be metrics satisfying the triangle inequality. The last section introduces statistical tests and estimators based on the minimal
$f$
-divergence with the empirical distribution achieved in the families of hypothetic distributions. For the Kullback divergence this leads to the classical likelihood ratio test and estimator.