ξi(t) =ξip+t Xi−t22ΓijkXjXk+O(‖tX‖3)
Usa-se o fato de que geodésicas são soluções do problema de valor inical:
$$ \begin{aligned} \ddot{\gamma}^i_{p,q}(t) & = - \Gamma^i_{jk}(t) \dot{\gamma}^j_{p,q}(t) \dot{\gamma}^k_{p,q}(t) \\ \dot{\gamma}_{p,q}(0) & = X(p,q) \end{aligned} $$
he canonical divergence D induces the metric g and the connections∇and∇∗. The same holds for the mean canonical divergence D∇mcd
if∇is integrable, then it is notgenerally true that X(q,p) =−gradqD∇mcd(p‖·)
mean canonical divergenceD∇mcd(p‖q):=12(D(p‖q) +D∗(q‖p))(64)which obviously satisfiesD(∇∗)mcd(p‖q) =D∇mcd(q‖p)
he energy of the geodesicγp,qas the symmetrized version of the canonical divergence:12(D(p‖q) +D(q‖p))=12∫10∥∥ ̇γp,q(t)∥∥2dt
Fazendo a mudança de variável $$ s \mapsto t(s) = 1- s $$ e usando o fato de que \( \gamma_{q,p}(s) = \gamma_{p,q}(1 - s) \), temos: $$ \begin{aligned} D(q||p) & := \int0^1 s ||\dot{\gamma}{q,p}(s)||^2 ds \ & = - \int1^0 (1 - t) ||\dot{\gamma}{p,q}(t)||^2 dt \ & = \int0^1 ||\dot{\gamma}{q,p}(t)||^2 dt - D(p||q) \end{aligned} $$
D(p‖q) =∫10t∥∥ ̇γp,q(t)∥∥2dt(61)whereγp,qdenotes the geodesic from p to q.
Até o momento, a conexão dual parece não desempenhar nenhum papel.
D(p‖q) =∫10(1−t)∥∥ ̇γq,p(t)∥∥2dt
inverse exponential map atγq,p(t)satisfiesXt(q,p)= (1−t) ̇γq,p(t)
$$ \tilde{\gamma}_{\gamma_{q,p}(t),p}(s) = \gamma_{q,p}(t + s(1-t)), s \in [0,1] $$
$$ \Longrightarrow X_t(q,p) := \dot{\tilde{\gamma}}_{\gamma_{q,p}(t),p}(s)\vert_{s=0} = (1 - t) \dot{\gamma}_{q,p}(t) $$
n-dimensional dual manifold(M,g,∇,∇∗). Consider a∇-geodesicγq,p:[0, 1]→Mconnectingqandp. We define a tangent vector fieldXt(p,q)along this geodesic:Xt(q,p):=X(γq,p(t),p)(52)Obviously,X0=X(q,p)(53)X1(q,p) =0(54)Definition 3.A canonical divergence from p to q is defined by the path integralD(p‖q) =∫10〈Xt(q,p), ̇γq,p(t)〉dt
Qual o papel da conexão dual?
∫10〈X(γ(t),p), ̇γ(t)〉dt=−∫10〈gradγ(t)Dp, ̇γ(t)〉dt=−∫10(dγ(t)Dp)( ̇γ(t))dt=−∫10d Dp◦γd t(t)dt=Dp(γ(0))−Dp(γ(1))=Dp(q)−Dp(p) =Dp(q) =D(p‖q)(13)In particular, we can apply this derivation to the geodesic connectingqandpeven when theintegrability ofXis not guaranteed and obtain the definition of a general canonical divergence
functionsDpsatisfying the condition of Equation (12) then they are uniqueup to a constant that can vary withp, and we can therefore assumeDp(p) =0
aRiemannian metricgonM. Given such a metric, we assumeintegrabilityofXand∇, respectively,in the sense that for allpthere exists a functionDpsatisfyingX(q,p) =−gradqDp
although being quite restrictive in general, thisproperty will be satisfied in our information-geometric context, wheregis given by the Fisher metricand∇is given by them- ande-connections and their convex combinations, theα-connections
pointspandq, one can interpret anyXwith expq(X) =pas a difference vectorXthattranslatesqtop
p−q=−gradqDp(9)Here, the gradient gradqis taken with respect to the canonical inner product onRn
De outra forma, podemos postular que a divergência canônica é a solução da edp: $$ D_p(q) = {1 \over 2 } |grad_q D_p|^2 $$
fixed pointp∈M, we want to define a vector fieldq7→X(q,p), at least in a neighbourhood ofp, thatcorresponds to the difference vector field
manifold is dually flat, a canonical divergence was introduced by Amari and Nagaoka [2], which isa Bregman divergence
a divergence exists for any such manifold. However, it isnot unique and there are infinitely many divergences that give the same geometrical structure
find a divergenceDwhich generates a given geometrical structure(M,g,∇,∇∗)
the coefficientsDΓijk(p) =−∂i∂j∂′kD(ξp‖ξq)∣∣q=p(5)DΓ∗ijk(p) =−∂′i∂′j∂kD(ξp‖ξq)∣∣∣q=p(6)define a pair of dual affine connectionsD∇andD∇∗[1]. The duality of the connections holds withrespect to the Riemannian metricDgin terms of the following condition:X〈Y,Z〉=〈D∇XY,Z〉+〈Y,D∇∗XZ〉(7)for all vector fieldsX,YandZ, where the brackets〈·,·〉denote the inner product with respect toDg
he coefficients of the Riemannian metric can be written asDgij(p) =−∂i∂′jD(ξp‖ξq)∣∣∣q=p=∂′i∂′jD(ξp‖ξq)∣∣∣q=p
When a coordinate systemξ:p7→ξp= (ξ1p, . . . ,ξnp)∈Rnis given inM, we pose one condition that, for two nearby pointsξpandξq=ξp+∆ξ,Dis expanded asD(p‖q) =12Dgij(p)∆ξi∆ξj+O(‖∆ξ‖3)(2)and(Dgij(p))ijis a positive definite matrix.
A divergence functionD(p‖q)is a differentiable real-valued function of two pointspandqin amanifoldM. It satisfies the non-negativity conditionD(p‖q)≥0(1)with equality if and only ifp=q.
A saturação (rigidez) da desigualdade é uma espécie de não-degenerescência da divergência.
Distribuições uniformes
Modelo estatístico não-regular.
um modelo estatístico regular de dimensãon−1
Não seria de dimensão n?