Page - 201 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Thefirst critiquearises fromrequirement (5). Fromtheviewpointoffiberbundlestherequirement(5) isuseless.ConsidertheCartesianproduct E=Θ×Ξ. That is thesamethingas the trivialfiberbundle E (θ,ξ)→π(θ,ξ)= θ∈Θ. ThereforePθ is therestrictionto thefiberEθ of the functionP. TheSecondCritique Thesecondcritiqueemerges fromtherequirement (1). This requirement (1) is toorestrictive. It excludesmanyinterestingcompactmanifoldssuchas flat tori, euclideansphere, compactLiegroups. TheThirdCritique From the viewpoint of the differential topology the requirement (6) may be damage to the topology ofΘ. When the Fisher information g is singular its kernel is in involution. Thus the topological-geometrical informationthatarecontainedingare transverse to thedistributionKer(g). IfKer(g) is completely integrable thentopologicalandgeometrical informationswhicharecontained ingare transversal to the foliationKer(g). SeePartAof thispaper. Thisends thecriticisms. Tomotivate for deleting the requirement (1)we construct a compact statisticalmodelwhich satisfiesallof therequirementsexcept therequirement (1). LetE bethe tangentbundleof thecircleS1.E is the trivial linebundle S1×R (θ,t)→ θ∈S1. Weconsider the fonctions f,FandPdefinedby f(θ,t)= [sin2( t2θ 1+ t2 )cos2( θ 4 )e−t 2 + π e2 t2], F(θ)= ∫ +∞ −∞ e−f(θ,t)dt, P(θ,t)= e−f(θ,t) F(θ) . The functionP(θ,t)has the followingproperties (i) (i) :P(θ,t) is smooth, (ii) P(0,t)=P(2π,t) ∀t∈R, (iii) the ddθ commuteswith ∫ R , (iv) P(θ,t)≤1 ∀(θ,t)∈S1×R, (v) if0< θ,θ∗<2π thenPθ=P∗θ if andonly ifθ= θ ∗, (vi) ∫+∞ −∞ P(θ,t)dt=1. Thesepropertiesshowthat there isaonetoonecorrespondencebetweenthecircleS1 andasubset of probabilitydensities inR. ThusS1 is a compact 1-dimensionalmanifoldof probabilities in the measurableset (R,β(R)).Hereβ(R) is the familyofBorel subsetsofR. So (S1,P) isacompactparametricmodel for (R,β(R)). 201