Page - 51 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 tobedefinedasahessianofpartition function logarithm gβ =−∂2Φ∂β2 = ∂2logψΩ ∂β2 as in classical informationgeometry.Wethenestablishtheequalityof twoterms, thefirstonegivenbySouriau’s definitionfromLiegroupcocycleΘandparameterizedby“geometricheat”Q (elementofdual Lie algebra) and“geometric temperature” β (element ofLie algebra) and the secondone, the hessianof thecharacteristic functionΦ(β)=−logψΩ(β)withrespect to thevariableβ: gβ([β,Z1] , [β,Z2])= 〈Θ(Z1) , [β,Z2]〉+〈Q, [Z1, [β,Z2]]〉= ∂ 2logψΩ ∂β2 (1) AdualSouriau-Fishermetric, theinverseof this lastone,couldbealsoelaboratedwiththehessian of “geometric entropy” s(Q)with respect to the variableQ: ∂ 2s(Q) ∂Q2 For themaximumentropy density (Gibbsdensity), the followingthree termscoincide: ∂ 2logψΩ ∂β2 thatdescribes theconvexityof the log-likelihoodfunction, I(β)=−E [ ∂2logpβ(ξ) ∂β2 ] theFishermetric thatdescribes thecovariance of the log-likelihoodgradient,whereas I(β)=E [ (ξ−Q)(ξ−Q)T ] =Var(ξ) thatdescribes the covarianceof theobservables. • ThisSouriau-Fishermetric isalso identifiedtobeproportional to thefirstderivativeof theheat gβ=−∂Q∂β , andthencomparablebyanalogytogeometric“specificheat”or“calorificcapacity”. • We observe that the Souriau metric is invariant with respect to the action of the group I ( Adg(β) ) = I(β), due to the fact that the characteristic functionΦ(β) after the actionof the group is linearly dependent to β. As the Fishermetric is proportional to the hessian of the characteristic function,wehavethe following invariance: I ( Adg(β) ) =−∂ 2(Φ−〈θ(g−1) ,β〉) ∂β2 =−∂ 2Φ ∂β2 = I(β) (2) • Wehaveproposed,basedonSouriau’sLiegroupmodelandonanalogywithmechanicalvariables, a variational principle of thermodynamics deduced fromPoincaré-Cartan integral invariant. Thevariationalprincipleholdsong theLiealgebra, forvariationsδβ= . η+[β,η],whereη(t) is anarbitrarypath thatvanishesat theendpoints,η(a)=η(b)=0: δ t1 t0 Φ(β(t)) ·dt=0 (3) where the Poincaré-Cartan integral invariant Ca Φ(β) ·dt = Cb Φ(β) ·dt is definedwithΦ(β), theMassieucharacteristic function,withthe1-formω=Φ(β)·dt=(〈Q,β〉−s)·dt=〈Q,(β·dt)〉−s·dt • WehavededucedEuler-Poincaréequations for theSouriaumodel: dQ dt =ad ∗ βQand ⎧⎨⎩ s(Q)=〈β,Q〉−Φ(β)β= ∂s(Q)∂Q ∈g ,Q= ∂Φ(β)∂β ∈g∗ and ddt ( Ad∗gQ ) =0 with { g∗ : dualLiealgebra ad∗XY: Coadjointoperator (4) whereQ is theSouriaugeometricheat (elementofdualLiealgebra)andβ is theSouriaugeometric temperature (elementof theLiealgebra). Thesecondequation is linkedto theresultofSouriau basedon themomentmap that a symplecticmanifold is alwaysacoadjointorbit, affineof its groupofHamiltoniantransformations (asymplecticmanifoldhomogeneousunder theactionofa Liegroup, is isomorphic,uptoacovering, toacoadjointorbit; symplectic leavesare theorbitsof theaffineactionthatmakes themomentmapequivariant). 51