Page - 111 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 theactiona is an inhomogeneous1-cocycle: ∀n>0, let the set of all functions fromGn toE∗, (Gn,E∗) called inhomogenesousn-cochains, thenwe candefine theoperators dn : (Gn,E∗)→ (Gn+1,E∗) by: dnF(g1, · · · ,gn+1)= g1.F(g2, · · · ,gn+1)+ n ∑ i=1 (−1)i F(g1,g2, · · · ,gigi+1, · · · ,gn) +(−1)n+1F(g1,g2, · · · ,gn) (C12) LetZn(G,E∗)=Ker(dn) ,B(G,E∗)= Im ( dn−1 ) ,withZn inhomogneousn-cocycles, thequotient: Hn(G,E∗)=Zn(G,E∗)/Bn(G,E∗) (C13) is theCohomologygroupofGwithvalue inE∗.Wehave: d0 :E∗→ (G,E∗) x → (g → g ·x−x) (C14) Z0={x∈E∗;g ·x= x,∀g∈G} (C15) d1 : (G,E∗)→ (G2,E∗) F → d1F , d1F(g1,g2)= g1 ·F(g2)−F(g1g2)+F(g1) (C16) Z1= { F∈ (G,E∗) ;F(g1g2)= g1 ·F(g2)+F(g1),∀(g1,g2)∈G2 } (C17) B1={F∈ (G,E∗) ;∃x∈E∗,F(g)= g ·x−x} (C18) WhentheCohomologygroupH1(G,E∗)=0 then: Z1(G,E∗)=B1(G,E∗) (C19) Then if F=F(μ) is anexponential family invariantbyG,μverifies: ∀g∈G,g(μ)(dx)= e〈c,x〉−〈c,g−1x〉+b(g)μ(dx) (C20) ∀g∈G,g ( e〈c,x〉μ(dx) ) = eb(g)e〈c,x〉μ(dx)withμ0(dx)= e〈c,x〉μ(dx) (C21) Forall compactgroup,H1(G,E∗)=0andwecanexpressa: A :G→GA(E) g →Ag , Ag(θ)= tg−1θ+a(g) (C22) ∀(g,g′)∈G2,Agg′=AgAg′ A(G)compactsub−groupofGA(E) (C23) ∃fixedpoint⇒∀g∈G,Ag(c)= tg−1c+a(g)= c⇒ a(g)= ( Id− tg−1 ) c (C24) References 1. Bernard, C. Introduction à l’Étudede laMédecine Expérimentale. Available online: http://classiques. uqac.ca/classiques/bernard_claude/intro_etude_medecine_exp/intro_medecine_exper.pdf (accessedon 17October2016). 2. Thom,R.Logos etThéoriedesCatastrophes;EditionsPatiño:Genève,Switzerland,1988. 111