The Δ2-Condition and ϕ-Families of Probability Distributions

Abstract

In this paper, we provide some results related to the ∆ 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 ∆ 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.

Geometry Induced by a Generalization of Rényi Divergence

“…This generalization also depends on a parameter α ∈ [−1, 1]; for α = ±1, it is defined as a limit. Supposing that the underlying ϕ-function is continuously differentiable, we show that this limit exists and results in the ϕ-divergence [12]. In what follows, all probability distributions are assumed to have positive density.…”

“…These conditions appeared first at [12] where the authors constructed non-parametric ϕ-families of probability distributions. We remark that if T is finite, condition (a3) is always satisfied.…”

“…An example of great relevance is the exponential function ϕ(u) = exp(u), which satisfies conditions (a1)-(a3) with u 0 = 1 T . Another example of ϕ-function is the Kaniadakis' κ-exponential [12,27,28].…”

“…In fact, ϕ-functions could be defined directly in terms of (a3'), without mentioning (a3). We chose to begin with (a3) because this condition appeared initially in [12]. Not all functions ϕ : R → (0, ∞), for which conditions (a1) and (a2) hold, satisfy condition (a3).…”

“…In particular, κ-exponential models and q-exponential families are investigated in [9,10]. Non-parametric (or infinite-dimensional) ϕ-families were introduced by the authors in [11,12], which generalize exponential families in the non-parametric setting [13][14][15][16]. Based on the similarity between exponential and ϕ-families, we defined the so-called ϕ-divergence, with respect to which the Kullback-Leibler divergence is a particular case.…”