Page - 210 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 is amorphismof [E,π,M,D,p] in [E∗,π∗,M∗,D∗,p∗] if p∗◦Ψ= p· AComment. Let [E,π,M,D]beanobjectof thecategoryFB(Γ,Ξ). LetGbethegroupof isomorphismsof [E,π,M,D]. IfMisfinitedimensional thenG isafinitedimensionalLiegroup. ThegroupGacts in the categoryMwhose objects areprobabilitydensities in [E,π,M,D, ]. Definition52. Theorbit space space m= M G iscalled themoduli spaceofM. AComment. Every trivializationof M=[E,π,M,D,p] is a statisticalmodel in the classical sense [18,22]. Sowehave takenupthe challenge1. 8.3.4. TwoAlternativeDefinitions Weintroduce twootherpresentationsof thecategoryGM(Ξ,Ω). Thosepresentationshighlight theconnectionwith thesearchesofMcCullaghandGromov. Thosepresentation isuseful inboth the theoretical statisticsandtheappliedstatistics [17,18,21,24,55,57,58]. Weconsider thecategoryMSEwhoseobjectsareaprobabilityspaces (Ξ,Ω,p). Definition 53. A morphism of a probability space (Ξ,Ω,p) in another probability space (Ξ∗,Ω∗,p∗) is ameasurablemapΨof (Ξ,Ω) in (Ξ∗,Ω∗) such that p= p∗◦Ψ· Remark4. Amorphismas in the last definition has a statistical nature. An isomorphisms of (Ξ,Ω,p) on (Ξ∗,Ω∗,p∗) is ansufficient statistic. ThecategoryMSE isuseful for introducing twovariantdescriptionsof the categoryGM(Ξ,Ω). Definition54. Weuse thepreviousnotation. (1) Astatisticalmodel is a locally trivialfiberbundleovera locallyflatmanifold π :E→M. Thefibersofπ areprobability spaces. (2) The functor [E,p]→ [M,D] is calledaMSE-fibration. The category of MSE-fibrations is denoted by FB(MSE). The morphisms the category GM(Ξ,Ω)arecalledMSE-morphisms. 210