Page - 319 - in Differential Geometrical Theory of Statistics

Entropy2017,19, 7 Next, let us consider the standard Fishermetric and the standard cubic form. Suppose that S := {p(x;θ)} is a statisticalmodel. Set pθ := p(x;θ), for simplicity. Wedefine the (standard)Fisher metric gF onSqby gFij(θ) := ∫ Ω (∂i lnpθ)(∂j lnpθ)pθdx, andthe (standard) cubic formor theAmari–ChentsovvectorfieldCF by CFijk(θ) := ∫ Ω (∂i lnpθ)(∂j lnpθ)(∂j lnpθ)pθdx. Fromsimilarargumentsof (11),wecandefineanα-connection∇(α)onSq, andwecanobtaina statisticalmanifoldstructure (Sq,∇(α),gF). In thiscase, (Sq,∇(α),gF) is calledan invariant statistical manifold [21,22]. A Fisher metric and a cubic form have the following representation using a sequence of escortdistributions, Theorem1. LetSq beaq-exponential family. For p(x;θ)∈Sq, suppose thatPq,(2)(x;θ)andPq,(3)(x;θ)are the secondandthe thirdescortdistributionof p(x;θ), respectively. ThentheFishermetric gF andthecubic form CF aregivenas follows: gFij(θ) = 1 q ∫ Ω (∂i lnq pθ)(∂j lnq pθ)Pq,(2)(x;θ)dx, (12) CFijk(θ) = 1 q(2q−1) ∫ Ω (∂i lnq pθ)(∂j lnq pθ)(∂k lnq pθ)Pq,(3)(x;θ)dx. (13) Proof. Differentiating theq-logarithm,wehave ∂i lnq pθ = ∂i ( p1−qθ −1 1−q ) = p−qθ ∂ip(θ)= p 1−q θ ∂i lnp(θ). Therefore,weobtain 1 q ∫ Ω (∂i lnq pθ)(∂j lnq pθ)Pq,(2)(x;θ)dx = ∫ Ω p1−qθ (∂i lnpθ)p 1−q θ (∂j lnpθ)p 2q−1 θ (x;θ)dx = ∫ Ω (∂i lnpθ)(∂j lnpθ)pθ(x;θ)dx = gFij(θ). Byasimilarargument,weobtain therepresentationforCF. Wedefinean α-divergenceD(α)with α = 1−2q anda q-relative entropy (or anormalizedTsallis relative entropy)DTq by D(1−2q)(p(x),r(x)) = 1 q Eq,p[lnq p(x)− lnq r(x)] = 1−∫Ωp(x)qr(x)1−qdx q(1−q) , (14) DTq(p(x),r(x)) = Eescq,p[lnq p(x)− lnq r(x)] = 1−∫Ωp(x)qr(x)1−qdx (1−q)Zq(p) , (15) respectively. It isknownthat theα-divergenceD(1−2q)(r,p) inducesastatisticalmanifoldstructure (Sq,gF,∇(2q−1)),wheregF is theFishermetriconSq and∇(2q−1) is theα-connectionwithα=2q−1, andtheq-relativeentropyDTq(r,p) induces (Sq,g,∇q(e)). 319