Page - 247 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 Proposition1. Let s∈Rr>0. Forξ∈P∗V, onehas: Δ−s(Js(ξ))−1=( r ∏ k=1 sskk )δ−s(ξ). (45) Proof. Weprove the statementby inductionon the rank. When r= 1, theequality (45) isverified directly. Indeed, the left-handsideof (45) is computedas ( s1ξ11) s1 = ss11 ξ −s1 11 . When r>1,assumethat (45)holds forasystemofrank r−1.Wededuce from(31), (33), (43), (44) andthe inductionhypothesis that (45)holds forξ∈P∗V∩Z0V. Therefore, (45)holds forallξ∈P∗V by (30), (34)and(42). Ingeneral, foranon-zerofunction f, thefunction 1f◦(∇log f)−1 iscalledthemultiplicativeLegendre transformof f. Thanks toTheorem5andProposition 1,we see that themultiplicative Legendre transformofΔ−s(x) isequal toδ−s(−ξ)on−P∗V uptoconstantmultiple.Asacorollary,wearriveat the followingresult. Theorem6. TheFenchel–Legendre transformof the convex function logΔ−s onPV is equal to the function logδ−s(−ξ)ofξ∈−P∗uptoconstantaddition. 6.ApplicationtoStatisticsandOptimization Take s∈Rr forwhich sk> qk/2 (k=1,. . . ,r).WedefineameasureρVs onP∗V by: ρVs (dξ) :=C−1V ΓV(s) −1δVs (ξ)ϕV(ξ)dξ (ξ∈P∗V). (46) Theorem4states that: ∫ P∗V e−(x|ξ)ρVs (dξ)=ΔV−s(x) (x∈PV). Then, we obtain the natural exponential family generated by ρVs , that is a family {μVs,x}x∈PV of probabilitymeasuresonP∗V givenby: μVs,x(dξ) :=ΔVs (x)e−(x|ξ)ρVs (dξ). In particular, when s = (n1α,n2α, . . . ,nrα) for sufficiently large α, we have μVs,x(dξ) = (detx)αe−(x|ξ)ρVs (dξ).WecallμVs,x theWishartdistributionsonP∗V ingeneral. Fromasampleξ0∈P∗V, letusestimate theparameterx∈PV insuchawaythat the likelihood functionΔVs (x)e−(x|ξ) attains itsmaximumat theestimatorx0. Then,wehavethe likelihoodequation ξ0=IVs (x0),whereasTheorem5givesauniquesolutionbyx0=JVs (ξ0). Thesameargument leadsus to the followingresult in semidefiniteprogramming. Forafixed ξ0∈P∗V andα>0,auniquesolutionx0 of theminimizationproblemof (x|ξ0)−αlogdetx subject to x∈PV=ZV∩Pn isgivenbyx0=JVs (ξ0),where s=(n1α, . . . ,nrα).Note thatJVs isarationalmap becauseδVs isaproductofpowersof rational functions. 7. SpecialCases 7.1.MatrixRealizationofHomogeneousCones Letusassumethat thesystemV={Vlk}1≤k<l≤r satisfiesnotonly theconditions (V1)and(V2), butalso the following: (V3)A∈Vlk,B∈Vkj⇒AB∈Vlj (1≤ j< k< l≤ r). 247