Page - 200 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Weintendto face the followingchallenges. Challenge1. Revisit the theoryofgeometric statisticalmodels formeasurablesets. Challenge2. TheSearch forageometric characteristic invariant for statisticalmodels.Werecall that suchan invariantwill encode thepoints themoduli spaceofmodels. Before continuingwe recall somedefinitions. Definition42. Ageometric invariantof amodel for (Ξ,Ω) is adatumwhich is invariantunder theactionof the symmetrygroupAut(Ξ,Ω). Theframeworkwhich isuseful for re-establishingthe theoryofstatisticalmodels is thecategory of locally trivialfiberbundles. Aswehavementionedtheneedfor introducinganewtheoryofstatisticalmodelemerges from somecriticisms.Werecall thedefinitionastatisticalmodel [18,22,24]. Definition43. Anm-dimensional statisticalmodel for ameasurable set (Ξ,Ω) is a pair (Θ,P)having the propertieswhich follow. (1) ThemanifoldΘ is anopensubset of them-dimensionalEuclideanspaceRm. (2) P is apositive realvalued function Θ×Ξ (θ,ξ)→P(θ,ξ)∈R subject to the requirementswhich follow. (3) The functionP(θ,ξ) isdifferentiablewith respect toθ∈Θ. (4) For everyfixedθ∈Θoneset Pθ=P(θ,−) then the triple (Ξ,Ω,Pθ) is aprobability space, viz ∫ Ξ Pθ(ξ)dξ=1 Furthermore theoperationofdifferentiation dθ= d dθ commuteswith theoperationof integration ∫ Ξ. (5) (Θ,P) is identifiable, viz forθ,θ∗ ∈Θ Pθ=Pθ∗ if andonly if θ= θ∗ (6) TheFisher information gθ(X,Y)= ∫ Ξ P(θ,ξ)[dθlog(P(θ,ξ))]⊗2(X,Y)dξ ispositivedefinite. SomeCriticisms. TheFirstCritique 200