^{*}

^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this paper, we introduce a geometry called _{X} of random variables. Amari’s

Geometric study of statistical estimation has opened up an interesting new area called the Information Geometry. Information geometry achieved a remarkable progress through the works of Amari [

A statistical manifold of probability distributions is equipped with a Riemannian metric and a pair of dual affine connections [

In this paper, Amari’s idea of using

In Section 3, a necessary and sufficient condition for two (

Let (_{X}^{1}, …, ^{n}^{n}^{1}, …, ^{n}

The tangent space to _{ξ}

Define _{i}l_{i}ℓ

Then there is a natural isomorphism between these two vector spaces _{ξ}

Obviously, a tangent vector
^{i}_{ξ}_{ξ}_{ξ}

Especially the inner product of the basis vectors _{i}_{j}

Note that ^{3} functions Γ_{ijk}

These functions Γ_{ijk}

∇ is called the 1−connection or the exponential connection.

Amari [

Using these, we can define ^{3} functions

These
^{α} on the statistical manifold

which are called

On a statistical manifold _{X}

Let _{X}_{i}F

It is clear that _{i}F_{i}℞_{i}F

Let the tangent space
_{ξ}_{ξ}

_{ξ}

For any
^{F}_{X}_{X}

We can use this _{X}

which induces an inner product on _{ξ}

^{F} on S is the Fisher information metric g

For any basis vectors _{i}, ∂_{j}_{ξ}

So the metric <, >^{F}_{X}

We can induce a connection on

Let

^{mn}_{mn}

From

Therefore

Hence we can write

Then we have the Christoffel symbols of the

and components of the

Let _{α}_{α}

The components

From _{α}

we have

Then we get

Hence

which are the components of the

_{ξ} as

^{F,G} reduces to the Fisher information metric. This is a more general way of defining Riemannian metrics and affine connections on a statistical manifold.

^{∗}

_{X} ^{F,G} and the^{H,G} are dual connections with respect to the

^{H,G}

∇^{F,G}^{H,G}

for any basis vectors _{i}, ∂_{j}, ∂_{k}_{ξ}

Then the condition (

Hence ∇^{F,G}^{H,G}^{H,G}

From Theorem 3.2, for ^{F}^{H}

Thus the ^{F}

That is, Amari’s 0–connection is the only self dual

So far, we have considered the statistical manifold

^{i}_{∂i}∂_{j}^{i}

^{1}, …, ^{n}^{n}^{i}

^{i}_{θ}

_{1}, …, _{n} on X such that

Let _{1}, …., _{n}_{i}_{i}_{j}_{ij}^{i}

Then we get

^{F},^{H}^{F},^{H} are the dual connections with respect to the Fisher information metric. Since F-connections are symmetric, the manifold S is flat with respect to^{F} iff S is flat with respect to^{H}. Thus if S is flat with respect to^{F}, then^{F},^{H}

For the statistical manifold ^{n}}

Let [^{i}_{j}_{ij}_{i}_{j}^{i}_{j}_{ijk}

Then the covariance of the metric

The components of the Fisher information metric with respect to the coordinate system [^{i}

Let
_{j}

Since

we can write

^{F} is covariant under re-parametrization.

Let the components of ∇^{F}^{i}_{j}_{ijk}

Let

The components of the ^{F}^{i}

The components of ∇^{F}_{j}

We can write

Then

Hence we get

Hence we showed that

Amari and Nagaoka [

^{n}^{n}

_{∗} is the push forward map associated with the map λ, which is defined by

Now we discuss the invariance properties of the

Consider a statistical manifold ^{n}

where

Let us denote log _{y}_{x}

Let

The Fisher information metric

which is the Fisher information metric on

The components of Amari’s

which are the components of Amari’s

Now we prove that

Let

and the components of the

Then by equating the components

Then it follows that the condition for

where k is a real constant.

Hence it follows from the Euler’s homogeneous function theorem that the function

Since

Therefore

Let

we get

which is nothing but Amari’s α–embeddings _{α}(

_{α}

The Fisher information metric and Amari’s _{X}

Amari’s

We are extremely thankful to Shun-ichi Amari for reading this article and encouraging our learning process. We would like to thank the reviewer who mentioned the references [

The authors contributed equally to the presented mathematical framework and the writing of the paper.

The authors declare no conflicts of interest.