Page - 110 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Let transformationkμ(θ)definedonΘ(u) interiorofDμ= { θ∈E∗,Lμ<∞ } : kμ(θ)= logLμ(θ) (C2) naturalexponential familiesaregivenby: F(μ)= { P(θ,μ)(dx)= e〈θ,x〉−kμ(θ)μ(dx),θ∈Θ(μ) } (C3) with injective function(domainofmeans): k′μ(θ)= E xP(θ,μ)μ(dx) (C4) the inverse function: ψμ :MF→Θ(μ)withMF= Im ( k′μ(Θ(μ)) ) (C5) andtheCovarianceoperator: VF(m)= k′′μ ( ψμ(m) ) = ( ψ′μ(m) )−1 , m∈MF (C6) MeasuregeneretadbyafamilyF is thengivenby: F(μ)=F(μ′)⇔∃(a,b)∈E∗×R,suchthatμ′(dx)= e〈a,x〉+bμ(dx) (C7) Let F an exponential family of E generated by μ and ϕ : x → gϕx+vϕ with gϕ ∈ GL(E) automorphismsofEandvϕ ∈ E, then the familyϕ(F)= {ϕ(P(θ,μ)) ,θ∈Θ(μ)} is anexponential famillyofEgeneratedbyϕ(μ) DefinitionC1.Anexponential familyF is invariantbyagroupG(affinegroupofE), if ∀ϕ∈G,ϕ(F)=F : ∀μ,F(ϕ(μ))=F(μ) (C8) (the contrarycouldbe false) ThenMurielCasalishas established the following theorem: TheoremC1(Casalis).Let F=F(μ)anexponential familyofEandGaffinegroupofE, thenF is invariant byG if andonly: ∃a :G→E∗,∃b :G→R, suchthat: ∀(ϕ,ϕ′)∈G2, ⎧⎨⎩ a(ϕϕ′)=tg−1ϕ a(ϕ′)+a(ϕ) b(ϕϕ′)= b(ϕ)+b(ϕ′)− 〈 a(ϕ′) ,g−1ϕ vϕ 〉 ∀ϕ∈G,ϕ(μ)(dx)= e〈a(ϕ),x〉+b(ϕ)μ(dx) (C9) WhenG isa linear subgroup, b is a character ofGanda couldbeobtainedby thehelpof cohomologyof Liegroups. IfwedefineactionofGonE∗ by: g ·x=tg−1x,g∈G,x∈E∗ (C10) It canbeverified that: a(g1g2)= g1 ·a(g2)+a(g1) (C11) 110