The \(\alpha\)-connections form the standard one-parameter family of affine connections in information geometry. They interpolate between mixture, Levi-Civita, and exponential geometry.
Definition by cubic tensor
The whole family is obtained by shifting Levi-Civita by the cubic tensor.
Definition. The \(\alpha\)-connection is defined by \[ \Gamma^{(\alpha)}_{ijk} =\Gamma^{(0)}_{ijk}-\frac{\alpha}{2}C_{ijk}. \] The dual of \(\nabla^{(\alpha)}\) is \(\nabla^{(-\alpha)}\).
Proposition. The duality relation follows from the symmetry of \(C\): \[ Xg(Y,Z)=g(\nabla_X^{(\alpha)}Y,Z)+g(Y,\nabla_X^{(-\alpha)}Z). \]
Proof. The Levi-Civita connection already satisfies metric compatibility. The two added cubic terms appear with opposite signs and cancel because \(C(X,Y,Z)=C(X,Z,Y)\).
Exponential and mixture endpoints
The values \(\alpha=1\) and \(\alpha=-1\) are statistically special.
Definition. With the common convention, \(\nabla^{(1)}\) is the exponential connection and \(\nabla^{(-1)}\) is the mixture connection. Exponential families are flat for \(\nabla^{(1)}\); mixture families are flat for \(\nabla^{(-1)}\).
Remark. Some authors reverse the signs. What matters is the dual pair and the identification of exponential versus mixture affine coordinates.
Geodesics
Different \(\alpha\) values declare different curves of distributions to be straight.
Proposition. On the positive simplex, \(\alpha=1\) geodesics are straight in log-probability coordinates up to normalization, while \(\alpha=-1\) geodesics are straight in probability coordinates.
Proof. The simplex has mixture coordinates \(p_i\), in which mixture geodesics are ordinary line segments. Exponential coordinates are log ratios, and exponential geodesics are ordinary line segments in those coordinates. These are precisely affine coordinates for the corresponding flat connections.
Levi-Civita midpoint
The Fisher-Rao geodesics arise at \(\alpha=0\), not at the exponential or mixture endpoints.
Remark. The \(\alpha=0\) geometry is metric and length-based. The \(\alpha=\pm1\) geometries are affine and projection-based. Both viewpoints are useful but answer different questions.
Definition and coordinate formula
Definition. The alpha-connection has lowered coefficients \[ \Gamma^{(\alpha)}_{ijk} =P_\theta\!\left[ \left\{\partial_i\partial_j\ell_\theta +\frac{1-\alpha}{2}\partial_i\ell_\theta\partial_j\ell_\theta \right\}\partial_k\ell_\theta \right], \] where \(\ell_\theta=\log p_\theta\). The conventions are \(\alpha=1\) for the exponential connection, \(\alpha=-1\) for the mixture connection, and \(\alpha=0\) for Levi--Civita.
Proposition. \[ \Gamma^{(\alpha)}_{ijk} =\Gamma^{(0)}_{ijk}-\frac{\alpha}{2}C_{ijk}, \qquad (\nabla^{(\alpha)})^*=\nabla^{(-\alpha)}. \]
Proof. Subtract the defining formulas for \(\alpha\) and \(0\); the only change is \(-\alpha P(\ell_i\ell_j\ell_k)/2\). The metric-duality identity for the \(\pm\alpha\) coefficients follows by differentiating \(g_{jk}=P(\ell_j\ell_k)\) and using \(P\ell_i=0\).
Exponential and mixture geodesics
Theorem. In a full regular exponential family \[ p_\theta(x)=\exp\{\theta^\top T(x)-\psi(\theta)\}h(x), \] natural coordinates \(\theta\) are affine for \(\nabla^{(1)}\). Mean coordinates \(\eta=\nabla\psi(\theta)=E_\theta T\) are affine for \(\nabla^{(-1)}\).
Proof. \(\partial_i\partial_j\ell_\theta=-\partial_{ij}\psi\) is independent of \(x\), so \[ \Gamma^{(1)}_{ijk} =-\partial_{ij}\psi\,P_\theta(\partial_k\ell_\theta)=0. \] Thus \(\theta\) is exponential-affine. Duality of the Hessian metric \(g=\nabla^2\psi\) implies that its Legendre-dual coordinates \(\eta=\nabla\psi\) make the mixture connection coefficients vanish.
On the full simplex, the two corresponding paths are \[ p_t^{(m)}=(1-t)p_0+tp_1, \qquad p_t^{(e)}(x) =\frac{p_0(x)^{1-t}p_1(x)^t} {\int p_0^{1-t}p_1^t\,\mathrm{d}\mu}. \] They are affine geodesics of different connections and need not minimize Fisher--Rao length.
Alpha-embeddings
For \(\alpha\ne1\), the coordinatewise map \[ L_\alpha(p)(x)=\frac{2}{1-\alpha} p(x)^{(1-\alpha)/2} \] turns alpha-geodesic calculations into affine geometry in a function space, subject to renormalization. The limit \(\alpha\to1\) is \(\log p\) up to an irrelevant divergent constant. These embeddings interpolate between square-root, mixture, and log-density representations.
Flatness is model dependent
Exponential families are \(e\)-flat, and mixture families are \(m\)-flat, but arbitrary curved submodels need not be flat for either connection. A one-dimensional connection is always locally flat, while in dimension at least two curvature becomes an obstruction to global affine coordinates. Statistical curvature measures the second-order departure of a curved family from its ambient exponential family.