Page - 227 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 isdefinedby fx(h)= f(h ·x). LetL∞H(M)bethesetof f ∗ ∈C∞(M)suchthat f∗x ∈L∞(H), viz sup [h∈H] |fx(h)|<∞ ∀x. NowEXP(L∞H(M)stands for thesetof f ∗ ∈L∞H(M)suchthat exp(f∗x(h))≤μ(exp(f∗x)) ∀x. The functionPf∗(x,h) isdefinedby Pf∗(x,h)= exp(f∗(x ·h)) μ(exp(f∗x)) . Thepair (M,Pf∗) isaprobabilitydensity inH.Nowset f˜∗(x,h)= f∗(x ·h) Therefore thedatum [M×H,p1,M,∇,Pf∗] isastatisticalmodel for (H,P(H)).HereP(M) is the booleanalgebraofsubsetsofHand p1 is the trivialfibrationofM×HoverM. Example2:Geometry Wefocusonanexamplewhichplaysasignificant role inglobalanalysis (andgeometry) insome typeofboundeddomains [2,3]. Thisexamplerelates thegeometryofKoszulandSouriauLiegroups thermodynamics [4]andbibliographytherein. LetC⊂RmbeaconvexconeandletC∗be itsdual. Thecharacteristic functionofC isdefinedby C v→ ∫ C∗ exp(−<v,w∗>)dw∗. Thisgivesrise to the followingfunction C×C∗ (v,v∗)→P(v,v∗)= exp(−<v,v ∗>)∫ C∗ exp(−<v,w∗>)dw∗ So (C,P) isastatisticalmodel for (C∗,dw∗).Heredw∗ is thestandardBorelmeasure. StratifiedAnalyticRiemannianFoliations Reminder. Werecall that a (regular)Riemannian foliationMis a symmetric bilinear formg∈S2(M)having the followingproperties (1) rank(g)= constant, (2) LXg=0∀X∈G(Ker(g)). From(2)one easilydeduces thatKer(g) is in involution. By thevirtueofTheoremofFrobenius (1) and (2) imply thatKer(g) is completely integrable. Inthecategoryofdifferentiablemanifolds,notall involutivesingulardistributionsarecompletely integrable. Nevertheless, that is true in the categoryof analyticmanifolds [62]. 227