Page - 195 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 whosedomain is a convexopensubsetU.Wewrite thematrixofFisher informationg in thebasis{ ∂ ∂θj } , namely g=∑gijdθidθj. Here gij=g( ∂ ∂θi , ∂ ∂θj ). Theassumption δKVg=0 is equivalent to the system ∂gij ∂θk − ∂gkj ∂θi =0 for all i, j,k. Weuseanotationwhich isused in [52].Weconsider thedifferential1-forms hj=∑ i gijdθi. Everydifferential1-formhj isadeRhamcocycle. BytheLemmaofPoincaré theconvexopensetUsupports smooth functionsφj, j=: 1,...,mwhichhave the followingproperty dφj=hj. Weput ω=∑ j φjdθj. Thenwehave (δKVω)( ∂ ∂θi , ∂ ∂θj )=gij. Thus thedifferential1-form∑jφjdθj is deRhamclosed. SinceU is convex it supports a local smooth functionΨ such that dΨ=∑φjdθj. Soweget g( ∂ ∂θi , ∂ ∂θj )= ∂2ψ ∂θi∂θj . Tocontinuewefixθ0∈Uandweconsider the function θ→ a(θ) which isdefined inUby a(θ)= ∫ Ξ P(θ0,ξ)[ψ(θ)+ log(P(θ,ξ))]dξ. Nowrecall that the integration ∫ Ξ commuteswith thedifferentiation d dθ. Therefore,∀i, j≤dim(Θ)onehas ∂2a ∂θi∂θj (θ)= ∫ Ξ P(θ0,ξ) ∂2(ψ+ log(P)) ∂θi∂θj (θ,ξ)dξ. 195