arXiv subjects

Rui F. Vigelis

Publications and source records attributed to Rui F. Vigelis.

6 recordsLinked to original sources

arXiv2020

Conditions for the existence of a generalization of Rényi divergence

We give necessary and sufficient conditions for the existence of a generalization of Rényi divergence, which is defined in terms of a deformed exponential function. If the underlying measure $μ$ is non-atomic, we found that not all deformed exponential functions can be used in the generalization of Rényi divergence; a condition involving the deformed exponential function is provided. In the case $μ$ is purely atomic (the counting measure on the set of natural numbers), we show that any deformed exponential function can be used in the generalization.

cs.IT↗

arXiv2020

Properties of a Generalized Divergence Related to Tsallis Relative Entropy

In this paper, we investigate the partition inequality, joint convexity, and Pinsker's inequality, for a divergence that generalizes the Tsallis Relative Entropy and Kullback-Leibler divergence. The generalized divergence is defined in terms of a deformed exponential function, which replaces the Tsallis $q$-exponential. We also constructed a family of probability distributions related to the generalized divergence. We found necessary and sufficient conditions for the partition inequality to be satisfied. A sufficient condition for the joint convexity was established. We proved that the generalized divergence satisfies the partition inequality, and is jointly convex, if, and only if, it coincides with the Tsallis relative entropy. As an application of partition inequality, a criterion for the Pinsker's inequality was found.

cs.IT↗

arXiv2015

New Metric and Connections in Statistical Manifolds

We define a metric and a family of $α$-connections in statistical manifolds, based on $φ$-divergence, which emerges in the framework of $φ$-families of probability distributions. This metric and $α$-connections generalize the Fisher information metric and Amari's $α$-connections. We also investigate the parallel transport associated with the $α$-connection for $α=1$.

math.PR↗

arXiv2014

Smoothness of the Orlicz Norm in Musielak-Orlicz Function Spaces

In this paper, we present a characterization of support functionals and smooth points in $L_{0}^Φ$, the Musielak-Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of $L_{0}^Φ$ is also obtained. Some expressions involving the norms of functionals in $(L_{0}^Φ)^{*}$, the topological dual of $L_{0}^Φ$, are proved for arbitrary Musielak-Orlicz functions.

math.FA↗

arXiv2013

On $φ$-families of probability distributions

We generalize the exponential family of probability distributions. In our approach, the exponential function is replaced by a $φ$-function, resulting in a $φ$-family of probability distributions. We show how $φ$-families are constructed. In a $φ$-family, the analogue of the cumulant-generating function is a normalizing function. We define the $φ$-divergence as the Bregman divergence associated to the normalizing function, providing a generalization of the Kullback-Leibler divergence. A formula for the $φ$-divergence where the $φ$-function is the Kaniadakis' $κ$-exponential function is derived.

math.PR↗

arXiv2013

The $Δ_2$-condition and $φ$-families of probability distributions

In this paper, we provide some results related to the $Δ_2$-condition of Musielak-Orlicz functions and $φ$-families of probability distributions, which are modeled on Musielak-Orlicz spaces. We show that if two $φ$-families are modeled on Musielak-Orlicz spaces generated by Musielak-Orlicz functions satisfying the $Δ_{2}$-condition, then these $φ$-families are equal as sets. We also investigate the behavior of the normalizing function near the boundary of the set on which a $φ$-family is defined.

math.PR↗