A statistical manifold is not only Riemannian. It carries a cubic tensor that measures the difference between dual affine connections and the Levi-Civita connection.
Amari-Chentsov tensor
The cubic tensor is the third-order score moment in a parametric model.
Definition. The Amari-Chentsov tensor is \[ C_{ijk}(\theta)=\mathbb E_\theta[s_is_js_k]. \] It is symmetric in all indices under regularity conditions.
Proposition. Under reparametrization, \(C_{ijk}\) transforms as a covariant 3-tensor.
Proof. Each score component transforms by the chain rule with one Jacobian factor. Multiplying three score components and taking expectation gives three Jacobian factors, which is exactly the covariant tensor transformation law.
Difference of dual connections
The cubic tensor records how far the exponential and mixture connections are from Levi-Civita.
Definition. For a statistical manifold, the \(\alpha\)-connections satisfy \[ \Gamma^{(\alpha)}_{ijk} =\Gamma^{(0)}_{ijk}-\frac{\alpha}{2}C_{ijk}, \] where \(\Gamma_{ijk}=g_{k\ell}\Gamma^\ell_{ij}\).
Remark. Here \(\alpha=0\) is Levi-Civita, \(\alpha=1\) is the exponential connection, and \(\alpha=-1\) is the mixture connection, up to convention.
Symmetry and torsion
Because the cubic tensor is symmetric, the \(\alpha\)-connections remain torsion-free.
Proposition. If \(\nabla^{(0)}\) is torsion-free and \(C_{ijk}\) is symmetric in \(i,j\), then every \(\nabla^{(\alpha)}\) is torsion-free.
Proof. Torsion in coordinates is the antisymmetric part of \(\Gamma^k_{ij}\) in \(i,j\). The Levi-Civita part has no antisymmetric component, and the added tensor term is symmetric in \(i,j\), so the antisymmetric part remains zero.
Skewness of the model
The cubic tensor captures third-order asymmetry of the model around a distribution.
Remark. For normal location families the cubic tensor vanishes in the location direction; for exponential families it encodes third cumulants of sufficient statistics.
The Amari--Chentsov tensor
Definition. For tangent vectors \(u,v,w\) with score representatives \(\ell_u,\ell_v,\ell_w\), define \[ C_\theta(u,v,w) =P_\theta(\ell_u\ell_v\ell_w). \] In coordinates, \(C_{ijk}=P_\theta(\ell_i\ell_j\ell_k)\). It is a symmetric covariant three-tensor.
Verification. Multilinearity and symmetry follow from multiplication. Under \(\eta=\eta(\theta)\), each score contributes one Jacobian factor: \[ C^{(\eta)}_{abc} =\frac{\partial\theta^i}{\partial\eta^a} \frac{\partial\theta^j}{\partial\eta^b} \frac{\partial\theta^k}{\partial\eta^c}C^{(\theta)}_{ijk}, \] which is the tensor transformation law.
Third-order information and skewness
The Fisher metric captures second-order separation of nearby laws. The cubic tensor records the leading asymmetry between opposite directions. Under smooth regularity, \[ D_{\mathrm{KL}}(P_\theta\|P_{\theta+h}) =\frac12g_{ij}h^ih^j +\frac16A_{ijk}h^ih^jh^k+O(\|h\|^4), \] where \(A\) combines derivatives of \(g\) and \(C\); reversing KL changes the odd-order part. Thus \(C\) is not itself a scalar skewness coefficient, but it is the intrinsic tensor governing the difference between dual third-order geometries.
Example (Bernoulli). With parameter \(p\), the score is \((X-p)/\{p(1-p)\}\), so \[ C_{ppp} =\frac{1-2p}{p^2(1-p)^2}. \] It vanishes at \(p=1/2\), where the binary model is locally symmetric, and changes sign across that point.
Example (normal location). In the fixed-variance normal location family, the score is centered Gaussian and all third moments vanish. Hence \(C=0\), the exponential and mixture connections coincide with Levi--Civita on this one-dimensional submanifold, and its information geometry is Euclidean.
Connections encoded by the cubic tensor
For the standard statistical dual pair, \[ \nabla^{(\alpha)} =\nabla^{(0)}-\frac{\alpha}{2}C^\sharp, \qquad g(C^\sharp(X,Y),Z)=C(X,Y,Z). \] Consequently, \[ \nabla^{(\alpha)}-\nabla^{(-\alpha)} =-\alpha C^\sharp. \] The metric and cubic tensor together determine the whole one-parameter family of alpha-connections. A statistical manifold may therefore be defined abstractly as a Riemannian manifold equipped with a symmetric cubic tensor, or equivalently with a torsion-free dual pair.