Page - 281 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 407 Proposition2. Inaϕ-familyFp, thegeneralizationofRényidivergence forα∈ (−1,1) canbe expressed in termsof thenormalizing functionψas follows: D(α)ϕ (pθ ‖ pϑ)= 21+αψ(θ)+ 2 1−αψ(ϑ)− 4 1−α2ψ (1−α 2 θ+ 1+α 2 ϑ ) , (29) for allθ,ϑ∈Θ. Proof. Recall thedefinitionofκ(α)as therealnumber forwhich∫ T ϕ (1−α 2 ϕ−1(pθ)+ 1+α 2 ϕ−1(pϑ)+κ(α)u0 ) dμ=1. Usingexpression(28) forprobabilitydistributions inFp,wecanwrite 1−α 2 ϕ−1(pθ)+ 1+α 2 ϕ−1(pϑ)+κ(α)u0 = c+ n ∑ i=1 (1−α 2 θi+ 1+α 2 ϑi ) ui− (1−α 2 ψ(θ)+ 1+α 2 ψ(ϑ)−κ(α) ) u0 = c+ n ∑ i=1 (1−α 2 θi+ 1+α 2 ϑi ) ui−ψ (1−α 2 θ+ 1+α 2 ϑ ) u0. The lastequality isaconsequenceof thedomainΘbeingconvex. Thus, it followsthat κ(α)= 1−α 2 ψ(θ)+ 1+α 2 ψ(ϑ)−ψ (1−α 2 θ+ 1+α 2 ϑ ) . By thedefinitionofD(α)ϕ (· ‖ ·),weget (29). Proposition3. Ina ϕ-familyFp, the ϕ-divergence is related to thenormalizing functionψby the equality Dϕ(pθ ‖ pϑ)=ψ(ϑ)−ψ(θ)−∇ψ(θ) ·(ϑ−θ), (30) for allθ,ϑ∈Θ. Proof. Toshow(30),weuse ∂ψ ∂θi (θ)= ∫ Tuiϕ ′(ϕ−1(pθ))dμ∫ Tu0ϕ ′(ϕ−1(pθ))dμ , which isaconsequenceof (Lemma10 in[12]). Inviewof (ϕ−1)′(u)=1/ϕ′(ϕ−1(u)), expression(13) with p= pθ andq= pϑ results in Dϕ(pθ ‖ pϑ)= ∫ T[ϕ −1(pθ)−ϕ−1(pϑ)]ϕ′(ϕ−1(pθ))dμ∫ Tu0ϕ ′(ϕ−1(pθ))dμ . (31) Inserting into (31) thedifference ϕ−1(pθ)−ϕ−1(pϑ)= ( c+ n ∑ i=1 θiui−ψ(θ)u0 ) − ( c+ n ∑ i=1 ϑiui−ψ(ϑ)u0 ) =ψ(ϑ)u0−ψ(θ)u0− n ∑ i=1 (ϑi−θi)ui, wegetexpression(30). 281