Page - 194 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 andasmooth function Θ θ→ψ(θ)∈R such that P(θ,ξ)= exp(C(ξ)+ m ∑ 1 Fj(ξ)θj−ψ(θ1,...,θm)). Theorem16. Let (Ξ,Ω)beameasurable set and let (Θ,P)beanm-dimensional statisticalmodel for (Ξ,Ω). TheFisher informationof (Θ,P) isdenotedbyg. The followingstatementsare equivalent. (1) There exists∇∈LF(Θ) such that δKVg=0, (2) Themodel (Θ,P) is anexponentialmodel. Demonstration. (2)⇒ (1). Weassumethat (2)holds. Thenwefixasystemofaffinecoordinate functions θ=(θ1,...,θm). Bythevirtueof (2)wehave P(θ,ξ)= exp(C(ξ)+ m ∑ 1 Fj(ξ)θj−ψ(θ)). Hereψ∈C∞(Θ)and (C,F)(ξ)=(C(ξ),F1(ξ),...,Fm(ξ))∈Rm+1. Therefore, onehas ∂2log(P(θ,ξ)) ∂θi∂θj =− ∂ 2ψ ∂θi∂θj . Therebyonecanwrite − ∫ Ξ P(θ,ξ) ∂2log(P(θ,ξ)) ∂θi∂θj = ∂2ψ(θ,ξ) ∂θi∂θj . This shows thatwehave g=δKV(dψ)∈B2KV(A,R). The implication (2)→ (1) isproved. (1)⇒ (2). Weusea strategysimilar to thatused in [52].Howeverourargumentsdonotdependonrank(g). Let∇∈LF(Θ)whoseKValgebra isdenotedbyA.Weassume g∈Z2KV(A,C∞(Θ)). Thuswehave δKVg=0. In (Θ,∇)wefixasystemof local affinecoordinate functions {θ1,...,θm} 194