Doubly Autoparallel Structure on the Probability Simplex

Abstract

On the probability simplex, we can consider the standard information geometric structure with the e- and m-affine connections mutually dual with respect to the Fisher metric. The geometry naturally defines submanifolds simultaneously autoparallel for the both affine connections, which we call doubly autoparallel submanifolds. In this note we discuss their several interesting common properties. Further, we algebraically characterize doubly autoparallel submanifolds on the probability simplex and give their classification.

A Complements for Proposition 1

We shortly summarize basic notions of the relation between submanifolds and affine connections and give an outline of the proof for Proposition 1.

For an n-dimensional manifold $\mathcal{M}$ equipped with an affine connection $\nabla$, we say that a submanifold $\mathcal{N}$ is autoparallel with respect to $\nabla$ when it holds that $\nabla _X Y \in \mathcal{X}(\mathcal{N})$ for arbitrary $X, Y \in \mathcal{X}(\mathcal{N})$. Since $\nabla ^{(\alpha )}$ is nothing but an affine combination of $\nabla =\nabla ^{(1)}$ and $\nabla ^*=\nabla ^{(-1)}$, we see that $\mathcal{N}$ is actually DA if $\mathcal{N}$ is autoparallel for arbitrary two connections $\nabla ^{(\alpha )}$ and $\nabla ^{(\alpha ')}$ with $\alpha \not =\alpha '$. Hence, the equivalence of the statements (1), (2) and (3) is straightforward.

If every $\nabla$-geodesic curve passing through a point $p \in \mathcal{N}$ that is tangent to $\mathcal{N}$ at p lies in $\mathcal{N}$, then $\mathcal{N}$ is called totally geodesic at p. If $\mathcal{N}$ is totally geodesic at every point of $\mathcal{N}$, then we say that $\mathcal{N}$ is a totally geodesic submanifold in $\mathcal{M}$. When $\nabla$ is torsion-free, a submanifold is totally geodesic if and only if it is autoparallel [11]. Since every $\nabla ^{(\alpha )}$ is torsion-free by the assumption, the equivalence of (3) and (4) holds.

When $\nabla$ is flat, there exists a coordinate system $(x^1, \ldots , x^n )$ of $\mathcal{M}$ satisfying $\nabla _{\partial /\partial x^i} \partial /\partial x^j=0$ for all i, j, which we call a $\nabla$-affine coordinate system. It is well known ([1, Theorem 1.1], for example) that a submanifold is $\nabla$-autoparallel if and only if it is expressed as an affine subspace with respect to $\nabla$-affine coordinate system. Then the equivalence of (1) and (5) immediately follows.

For the final statement a proof is given in [12] when $\mathcal{M}$ is the probability simplex. Since the proof for this general case is similar, it is omitted.