Page - 83 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Figure9.GenerationofKoszulelements fromCartan innerproduct. This informationgeometryhasbeen intensivelystudiedforstructuredmatrices [151–166]andin statistics [167]andis linkedto theseminalworkofSiegel [168]onsymmetricboundeddomains. Wecanapply thisKoszulgeometry frameworkforconesofsymmetricpositivedefinitematrices. Let the innerproduct 〈η,ξ〉=Tr(ηTξ) ,∀η,ξ∈Sym(n)givenbyCartan-Killingform,Ωbethesetof symmetricpositivedefinitematrices isanopenconvexconeandisself-dualΩ∗=Ω. 〈η,ξ〉=Tr(ηTξ) ,∀η,ξ∈Sym(n) ψΩ(β)= Ω∗ e−〈β,ξ〉dξ=det(β)−n+12 ψΩ(Id) ξˆ= ∂Φ(β) ∂β = ∂(−logψΩ(β)) ∂β = n+1 2 β−1 (158) pξˆ(ξ)= e −〈Θ−1(ξˆ),ξ〉+Φ(Θ−1(ξˆ)) =ψΩ(Id) · [ det ( αξˆ−1 )] ·e−Tr(αξˆ−1ξ) withα= n+1 2 (159) Wewill in the following illustrate informationgeometry formultivariateGaussiandensity [169]: pξˆ(ξ)= 1 (2π)n/2det(R)1/2 e− 1 2(z−m)TR−1(z−m) (160) Ifwedevelop: 1 2 (z−m)TR−1(z−m) = 1 2 [ zTR−1z−mTR−1z−zTR−1m+mTR−1m] = 1 2 zTR−1z−mTR−1z+ 12mTR−1m (161) Wecanwrite thedensityasaGibbsdensity: pξˆ(ξ)= 1 (2π)n/2det(R)1/2e 1 2m TR−1m e−[−mTR−1z+12zTR−1z] = 1 Z e−〈ξ,β〉 ξ= [ z zzT ] andβ= ⎡⎣ −R−1m1 2 R−1 ⎤⎦=[ a H ] with 〈ξ,β〉= aTz+zTHz=Tr[zaT+HTzzT] (162) 83