Rescaling Entropy and Divergence Rates

Loïck Lhote (1) , Valérie Girardin (2)

1 Equipe AMACC - Laboratoire GREYC - UMR6072
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

Based on rescaling by some suitable sequence instead of the number of time units, the usual notion of divergence rate is here extended to define and determine meaningful generalized divergence rates. Rescaling entropy rates appears as a special case. Suitable rescaling is naturally induced by the asymptotic behavior of the marginal divergences. Closed form formulas are obtained as soon as the marginal divergences behave like powers of some analytical functions. A wide class of countable Markov chains is proven to satisfy this property. Most divergence and entropy functionals defined in the literature are concerned, e.g., the classical Shannon, Kullback-Leibler, R´enyi, Tsallis. For illustration purposes, Ferreri or Basu-Harris-Hjort-Jones – among others – are also considered.

Mots clés: Kullback-Leibler divergence; Markov chain; Renyi entropy; Shannon entropy; Tsallis entropy; divergence rate; entropy functional; entropy rate

Domaines: Mathématiques [math]; Informatique [cs]; Théorie de l'information [cs.IT]

Loïck Lhote, Valérie Girardin. Rescaling Entropy and Divergence Rates. IEEE Transactions on Information Theory, 2015, 61 (11), pp.5868-5882. ⟨10.1109/TIT.2015.2476486⟩. ⟨hal-01196817⟩