Statistical Manifold

Statistical Manifold

Amari-Chentsov Tensor

Given a pair of conjugate connections and on , the Amari-Chentsov tensor is a -tensor defined byor in coordinate-free form,

Proposition

The Amari-Chentsov tensor is totally symmetric if the conjugate connections are torsion-free.

Proof Since both connections are torsion-free, it is easy to show that . By conjugation, we have Now consider Therefore since any symmetric group is generated by adjacent transpositions, the Amari-Chentsov tensor is totally symmetric.

Statistical Manifold

A statistical manifold is a manifold equipped with a metric tensor , a pair of torsion-free conjugate connections and corresponding totally symmetric Amari-Chentsov .

-Connections

-Connection

For any statistical manifold , we can define a family of connections called -connections, such that

Proposition

is torsion free for any .

Proof Choose some coordinate chart, then we can derive the coordinate form: where . Observe that they don’t change if swap indices and , thus they are torsion free by theorem.

Proposition

For any fixed , forms a pair of conjugate connection.

Proof The last equation holds since the Levi-Civita connection is self-conjugate.

Therefore for any , forms a statistical manifold with totally symmetric cubic tensor being defined correspondingly.

Proposition

The -connections can also be constructed directly from a pair of conjugate connections by taking the following weighted combination: