Page - 202 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 ADigression. Let { (Θj,Pj), j :=1,2 } be statisticalmodels formeasurable sets { (Ξj,Ωj), j :=1,2 } .Weput Θ=Θ1×Θ2, (Ξ,Ω)=(Ξ1×Ξ2,Ω1×Ω2), P=P1⊗P2. The functionP isdefined inΘ×Ξby P((θ1,θ2),(ξ1,ξ2))=P1(θ1,ξ1)P2(θ2,ξ2) The integrationonΞ isdefinedby∫ Ξ f((θ1,θ2),(ξ1,ξ2))d(ξ1,ξ2)= ∫ Ξ1×Ξ2 f((θ1,θ2),(ξ1,ξ2))dξ1dξ2. Thusweget ∫ Ξ P[(θ1,θ2),(ξ1,ξ2)]dξ1dξ2= ∫ Ξ1×Ξ2 P1(θ1,ξ1)P2(θ2,ξ2)dξ1dξ2=1. So (Θ,P) is a statisticalmodel for [Ξ1×Ξ2,Ω1×Ω2]. One is inposition toprove that everyEuclidean torusTm isa statisticalmodel for (Rm,β(Rm)). AnotherConstruction. Foreverypositive integermweconsiderpositive realnumbers α1<α2< ...<αm andthe real functionswhicharedefinedby fj(θ,t)= sin2( t2θ 1+ t2 )cos2( θ 4 )e−t2+αjt2 (θ,t)∈E, Fj(θ)= ∫ +∞ −∞ e−fj(θ,t)dt, Pj(θ,t)= e−fj(θ,t) Fj(θ) . Nowweconsider the tangentbundleof them-dimensionalflat torusTTm, T m=S1×S1× ...×S1. Let (θ,t)= [(θ1,t1),(θ2,t2),...,(θm,tm)]∈TTm. Weput F(θ)= ∫ Rm e−∑ m 1 fj(θj,tj)dt1dt2...dtm, P(θ,t)= e−∑ m 1 fj(θj,tj) F(θ) . The functionP(θ,t) satisfies the followingrequirements 202