One-dimensional exponential families with constant Hessian scalar curvature

TL;DR

This paper classifies one-dimensional exponential families with finite sample spaces that have constant Hessian scalar curvature, revealing a specific relationship between the curvature and integer parameters, and highlighting the binomial distribution's role.

Contribution

It provides a complete classification of 1D exponential families with constant Hessian scalar curvature, identifying the curvature values and the significance of the binomial distribution.

Findings

Hessian scalar curvature is quantized as 2/k for positive integers k

Binomial distribution plays a central role in the classification

Explicit characterization of exponential families with constant curvature

Abstract

We give a complete classification of 1-dimensional exponential families $\mathcal{E}$ defined over a finite space $\Omega=\{x_{0}, ...,x_{n}\}$ whose Hessian scalar curvature is constant. We observe an interesting phenomenon: if $\mathcal{E}$ has constant Hessian scalar curvature, say $\lambda$, then $\lambda=\tfrac{2}{k}$ for some positive integer $k\leq m$. We also discuss the central role played by the binomial distribution in this classification.

Equations

$$E_m/\sim \cong {\text{Graff}}_1({\mathbb{R}}^m).$$

$${\mathbb{R}}^{2n}\to {\mathbb{C}}^n,(x_1,...,x_n,y_1,...,y_n)\mapsto (x_1+i{y}_1,...,x_n+i{y}_n).$$

$$(J_{\alpha }{)}_p:=({\varphi }_{\alpha }^{-1}{)}_{{\ast }_{\varphi (p)}}\circ J_{{\mathbb{R}}^{2n}}\circ ({\varphi }_{\alpha }{)}_{{\ast }_p},$$

$$\left[\begin{array}{cc}0& -I_n\\ I_n& 0\end{array}\right].$$

$$N^J(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY],$$

$${\omega }_p(u,v):=g_p(Ju,v),(p\in M,u,v\in T_{p}M).$$

$$\omega =\sum _{k=1}^{n}d{x}_k\wedge d{y}_k,$$

$${\omega }_p(u,v):=-\text{det}(p,u,v)=-g_p(p,u\times v),$$

$$(\pi {|}_{S^{2n-1}}{)}^{\ast }g=j^{\ast }\text{Re}⟨\cdot ,\cdot ⟩\text{and}(\pi {|}_{S^{2n-1}}{)}^{\ast }\omega =j^{\ast }\text{Im}⟨\cdot ,\cdot ⟩,$$

$$\displaystyle\nabla_{X}Y=\sum_{k=1}^{n}\bigg{(}X(Y^{k})+\sum_{i,j=1}^{n}X^{i}Y^{j}\Gamma_{ij}^{k}\bigg{)}\dfrac{\partial}{\partial x_{k}},$$

$${\nabla }_{\frac{\partial }{\partial x_i}}\frac{\partial }{\partial x_j}=\sum _{k=1}^n{\Gamma }_{ij}^k\frac{\partial }{\partial x_k}\text{for}i,j=1,...,n.$$

$$\displaystyle\widetilde{\varphi}\bigg{(}\sum_{i=1}^{n}u_{i}\dfrac{\partial}{\partial x_{i}}\bigg{|}_{p}\bigg{)}=(x_{1}(p),...,x_{n}(p),u_{1},...,u_{n}).$$

$$\displaystyle K_{u}\bigg{(}\dfrac{\partial}{\partial q_{a}}\bigg{|}_{u}\bigg{)}:=\sum_{k,j=1}^{n}\Gamma_{aj}^{k}(p)u_{j}\dfrac{\partial}{\partial x_{k}}\bigg{|}_{p}\,\,\,\,\,\,\,\,\,\,\textup{for}\,\,a=1,...,n,$$

$$\displaystyle K_{u}\bigg{(}\dfrac{\partial}{\partial r_{a}}\bigg{|}_{u}\bigg{)}:=\dfrac{\partial}{\partial x_{a}}\bigg{|}_{p}\,\,\,\,\,\,\,\,\,\,\textup{for}\,\,a=1,...,n.$$

$$\displaystyle Y_{*_{p}}X_{p}=\sum_{a=1}^{n}\bigg{(}X^{a}(p)\dfrac{\partial}{\partial q_{a}}\bigg{|}_{u}+X_{p}(Y^{a})\dfrac{\partial}{\partial r_{a}}\bigg{|}_{u}\bigg{)}$$

$$\displaystyle K_{u}(Y_{*_{p}}X_{p})=\sum_{a=1}^{n}\bigg{(}X^{a}(p)\sum_{k,j=1}^{n}\Gamma_{aj}^{k}(p)Y^{j}(p)\dfrac{\partial}{\partial x_{k}}\bigg{|}_{p}+X_{p}(Y^{a})\dfrac{\partial}{\partial x_{a}}\bigg{|}_{p}\bigg{)},$$

$$K:TTM\to TM,$$

$$A\mapsto ({\pi }_{{\ast }_u}A,KA),$$

$$\displaystyle g_{u}\big{(}\big{(}v,w\big{)},\big{(}\overline{v},\overline{w}\big{)}\big{)}\quad:=$$

$$\displaystyle\omega_{u}\big{(}\big{(}v,w\big{)},\big{(}\overline{v},\overline{w}\big{)}\big{)}\quad:=$$

$$\displaystyle J_{u}\big{(}\big{(}v,w\big{)}\big{)}\quad:=$$

$$Xg(Y,Z)=g({\nabla }_{X}Y,Z)+g(Y,{\nabla }_X^{\ast }Z)$$

$$T(X,Y)$$

$$R(X,Y)Z$$

$${\nabla }_{\frac{\partial }{\partial x_i}}\frac{\partial }{\partial x_j}=0$$

$$g=\left[\begin{array}{cc}{h}_{ij}& 0\\ 0& h_{ij}\end{array}\right],J=\left[\begin{array}{cc}0& -I_n\\ I_n& 0\end{array}\right],\omega =\left[\begin{array}{cc}0& h_{ij}\\ -h_{ij}& 0\end{array}\right],$$

$$\text{Ric}(X,Y):=\text{trace}\{Z\mapsto R(Z,X)Y\},$$

$$Z_{\alpha}:=\frac{\partial}{\partial z_{\alpha}}=\frac{1}{2}\biggl{(}\frac{\partial}{\partial x_{\alpha}}-i\frac{\partial}{\partial y_{\alpha}}\biggr{)}\qquad\textup{and}\qquad\overline{Z}_{\alpha}:=\frac{\partial}{\partial\overline{z}_{\alpha}}=\frac{1}{2}\biggl{(}\frac{\partial}{\partial x_{\alpha}}+i\frac{\partial}{\partial y_{\alpha}}\biggr{)},$$

$${\text{Ric}}_{\alpha \beta }={\text{Ric}}_{\overline{\alpha }\overline{\beta }}\equiv 0,{\text{Ric}}_{\alpha \overline{\beta }}=\overline{{\text{Ric}}_{\overline{\alpha }\beta }}\text{and}{\text{Ric}}_{\alpha \overline{\beta }}=-\frac{{\partial }^2\ln \det (G)}{\partial z_{\alpha }\partial {\overline{z}}_{\beta }},$$

$$g_{\alpha\overline{\beta}}=\frac{1}{2}h_{\alpha\beta}\circ\pi\qquad\textup{and}\qquad\textup{Ric}_{\alpha\overline{\beta}}=-\frac{1}{4}\biggl{(}\frac{{\displaystyle\partial^{2}\ln d}}{\partial x_{\alpha}\partial x_{\beta}}\biggr{)}\circ\pi,$$

$$\displaystyle\dfrac{\partial}{\partial q_{k}}=\dfrac{\partial}{\partial z_{k}}+\frac{\partial}{\partial\overline{z}_{k}}\qquad\textup{and}\qquad\dfrac{\partial}{\partial r_{k}}=i\biggl{(}\frac{\partial}{\partial z_{k}}-\frac{\partial}{\partial\overline{z}_{k}}\biggr{)},$$

$$\displaystyle\textup{Ric}\bigl{(}q,r\bigr{)}=\begin{bmatrix}\beta_{\alpha\beta}&0\\ 0&\beta_{\alpha\beta}\end{bmatrix},\qquad\textup{where}\quad\>\beta_{\alpha\beta}=-\frac{1}{2}\frac{{\displaystyle\partial^{2}\ln d}}{\partial x_{\alpha}\partial x_{\beta}},$$

Full text

1 Introduction

Let $\Omega=\{x_{0},...,x_{m}\}$ be a finite set endowed with the counting measure and let $\mathcal{E}$ be a 1-dimensional exponential family defined over $\Omega$, with elements of the form $p(x;\theta)=\textup{exp}(C(x)+\theta F(x)-\psi(\theta))$, where $C,F:\Omega\to\mathbb{R}$ are functions, $\theta\in\mathbb{R}$ and $\psi:\mathbb{R}\to\mathbb{R}$. We denote by $h_{F}$, $\nabla^{(e)}$ and $\nabla^{(m)}$, the Fisher metric, exponential connection and mixture connection, respectively.

As it is well-known, the dualistic structure $(h_{F},\nabla^{(e)},\nabla^{(m)})$ is dually flat [AN00]. Therefore, the tangent bundle $T\mathcal{E}$ is naturally a Kähler manifold of real dimension 2 [Mol13, Shi07]. Let $\textup{Scal}:T\mathcal{E}\to\mathbb{R}$ be the corresponding scalar curvature.

In this paper, we classify all 1-dimensional exponential families, as described above, for which Scal is constant (see Theorem 7.9). Our proof is based on the particularly simple expression for the Ricci tensor in complex coordinates (since $T\mathcal{E}$ is Kähler), which implies that Scal factorizes as $\textup{Scal}=S\circ\pi$, where $\pi:T\mathcal{E}\to\mathcal{E}$ is the canonical projection and $S:\mathcal{E}\to\mathbb{R}$ is a globally well-defined function. Thus, solving the equation $\textup{Scal}\equiv\textup{constant}$ amounts to solve the simpler equation $S\equiv\textup{constant}$, which can be done by solving elementary differential equations in one variable.

An interesting consequence of the above classification is that if $T\mathcal{E}$ has constant scalar curvature, then $\textup{Scal}=\tfrac{2}{k}$ for some positive integer $k$ satisfying $1\leq k\leq m$ (see Corollary 7.10). For instance, if $\mathcal{E}=\mathcal{B}(n)$ is the set of binomial distributions defined over $\{0,1,...,n\}$, then $\textup{Scal}=\tfrac{2}{n}$.

The last section of the paper is devoted to analysing the “internal symmetries” of the problem, leading to a somewhat simpler reformulation of the classification discussed above that emphasizes the importance of the binomial distribution. For this purpose, we introduce an equivalence relation $\sim$ on the set $E_{m}$ of all 1-dimensional exponential families defined over the same set $\Omega=\{x_{0},...,x_{m}\}$ by declaring $\mathcal{E}\in E_{m}$ to be equivalent to $\mathcal{E}^{\prime}\in E_{m}$ if and only if they coincide as spaces of maps $\Omega\to\mathbb{R}$. In Proposition 8.4, we show that the set of equivalence classes is in one-to-one correspondence with the affine Grassmannian $\textup{Graff}_{1}(\mathbb{R}^{m})$ of 1-dimensional affine subspaces of $\mathbb{R}^{m}$, that is,

[TABLE]

Then, given $\mathcal{E}\in E_{m}$, we introduce the reduced exponential family of $\mathcal{E}$, denoted by $\mathcal{E}_{red}$ (see Definition 8.6). It is an exponential family defined over a finite set $\Omega_{red}$, with elements of the form $\textup{exp}\{C_{red}(x)+\theta F_{red}(x)-\psi_{red}(\theta)\}$. Its dualistic structure is isomorphic to that of $\mathcal{E}$, but in general $\Omega_{red}\neq\Omega$, and $F_{red}$ is always strictly increasing. Then we reformulate the classification given in Theorem 7.9 as follows (see Proposition 8.8). If $\mathcal{E}$ is a 1-dimensional exponential family defined over $\Omega=\{x_{0},...,x_{m}\}$, then $\textup{Scal}:T\mathcal{E}\to\mathbb{R}$ is constant if and only if $\mathcal{E}_{red}\sim\mathcal{B}(p)$, where $p+1$ is the cardinality of $\Omega_{red}$.

For the convenience of the reader, the paper gives a rather detailed discussion on the relation between Kähler geometry and statistics. The topics covered include: definition and examples of Kähler manifolds (Section 2), connections and connectors (Section 3), Dombrowki’s construction (Section 4), Ricci curvature (Section 5) and statistical manifolds (Section 6).

In Section 7, we classify all 1-dimensional exponential families $\mathcal{E}$ with constant scalar curvature on $T\mathcal{E}$ (Theorem 7.9).

In Section 8, we reformulate the classification result obtained in the preceding section by using equivalence classes and reduced exponential families (Proposition 8.8).

Notations. If $M$ is a manifold, then $\mathfrak{X}(M)$ will denote the space of vector fields on $M$ and $C^{\infty}(M)$ the space of smooth real-valued functions on $M$. $TM$ will denote the tangent bundle of $M$ and $TTM$ the tangent bundle of the tangent bundle of $M$ (hence if $\textup{dim}(M)=n$, then $\textup{dim}(TTM)=4n$). The derivative of a smooth map $f:M\to N$ between manifolds at a point $p\in M$ will be denoted by $f_{*_{p}}$.

2 Kähler manifolds

General references are [Bal06, Huy05, Mor07].

Definition 2.1**.**

A complex manifold of complex dimension $n$ is a Hausdorff topological space $M$ together with a family of maps $\phi_{\alpha}:U_{\alpha}\to\mathbb{C}^{n}$, $\alpha\in\>A$, where each $U_{\alpha}\subset M$ is an open set, such that:

• •

$\cup_{\alpha\in\>A}$ $U_{\alpha}=M$,

• •

$\phi_{\alpha}(U_{\alpha})$ is an open subset of $\mathbb{C}^{n}$ for all $\alpha\in A$,

• •

$\phi_{\alpha}$ is a homeomorphism onto its image for all $\alpha\in A$,

• •

$\phi_{\beta}\circ\phi_{\alpha}^{-1}:\phi_{\alpha}(U_{\alpha}\cap U_{\beta})\to\phi_{\beta}(U_{\alpha}\cap U_{\beta})$ is holomorphic for all $\alpha,\beta\in A$ (provided $U_{\alpha}\cap U_{\beta}\neq\emptyset$).

The family \mathcal{A}:=\big{\{}(U_{\alpha},\phi_{\alpha})\,|\,\alpha\in A\big{\}} is called a complex atlas.

Just as for smooth manifolds, one defines complex charts and complex coordinates on a complex manifold $M$.

In what follows, we will often identify $\mathbb{C}^{n}$ with $\mathbb{R}^{2n}$ via the map

[TABLE]

Upon this identification, every complex atlas of $M$ determines a smooth atlas of real dimension $2n$. Therefore, complex manifolds of complex dimension $n$ are naturally smooth manifolds of dimension $2n$.

Let $M$ be a complex manifold of complex dimension $n$ with complex atlas \mathcal{A}=\big{\{}(U_{\alpha},\phi_{\alpha})\,\big{|}\,\alpha\in A\big{\}}. For each $\alpha\in A$ and each $p\in U_{\alpha}$, define $(J_{\alpha})_{p}\,:\,T_{p}M\rightarrow T_{p}M$ by

[TABLE]

where $J_{\mathbb{R}^{2n}}\,:\,\mathbb{R}^{2n}\to\mathbb{R}^{2n}$ is the linear map whose matrix representation in the canonical basis is

[TABLE]

It is easy to check that if $p\in U_{\alpha}\cap U_{\beta}$, then $(J_{\alpha})_{p}=(J_{\beta})_{p}$. Thus we will use the notation $J_{p}$ instead of $(J_{\alpha})_{p}$. Letting $p\in M$ vary, we obtain a smooth tensor $J\,:\,TM\rightarrow TM$ on the smooth manifold $M$ that satisfies $J\circ J=-Id$.

The tensor $J$ is called the complex structure of the complex manifold $M$.

Definition 2.2**.**

Let $M$ be a smooth manifold. A smooth tensor $J\,:\,TM\rightarrow TM$ satisfying $J\circ J=-Id$ is called an almost complex structure.

Example 2.3**.**

The complex structure of a complex manifold is an almost complex structure.

An almost complex structure $J$ on a smooth manifold $M$ is said to be integrable if there exists a complex atlas on $M$ whose corresponding complex structure coincides with $J$. The Newlander-Nirenberg Theorem asserts that an almost complex structure $J$ is integrable if and only if the Nijenhuis tensor, defined by

[TABLE]

vanishes identically for all vector fields $X,Y$ on $M$ (see [NN57]).

Therefore, a complex manifold can be viewed as a pair $(M,J)$, where $M$ is a smooth manifold and $J\,:\,TM\rightarrow TM$ is an integrable almost complex structure.

Definition 2.4**.**

An almost Hermitian manifold is a triple $(M,g,J)$, where $M$ is a smooth manifold, $g$ is a Riemannian metric and $J$ is an almost complex structure such that $g_{p}(Ju,Jv)=g_{p}(u,v)$ for all $p\in M$ and all $u,v\in T_{p}M$.

If $(M,g,J)$ is an almost Hermitian manifold, we define a 2-form $\omega$ on $M$, called the fundamental form, by

[TABLE]

Definition 2.5**.**

A Kähler manifold is an almost Hermitian manifold $(M,g,J)$ satisfying the following analytical conditions:

1. (i)

the fundamental form $\omega$ is closed, that is, $d\omega=0$, 2. (ii)

$J$ is integrable.

Example 2.6**.**

The manifold $M=\mathbb{\mathbb{C}}^{n}\simeq\mathbb{R}^{2n}$ endowed with the Euclidean metric and the almost complex structure $J_{\mathbb{R}^{2n}}$ is a Kähler manifold, whose fundamental form is

[TABLE]

where $x_{1},...x_{n},y_{1},...,y_{n}$ are linear coordinates of $\mathbb{R}^{2n}$.

Example 2.7**.**

The unit sphere $S^{2}\subseteq\mathbb{R}^{3}$ endowed with the round metric $g$ induced by $\mathbb{R}^{3}$, and complex structure $J_{p}(u):=-p\times u$ (cross product) is a Kähler manifold with fundamental form $\omega$ given by

[TABLE]

where $p\in S^{2}$ and $u,v\in T_{p}S^{2}=\{\textup{plane orthogonal to}\,p\}$.

Example 2.8**.**

The complex projective space $\mathbb{P}(\mathbb{C}^{n})$ is the set of all complex lines in $\mathbb{C}^{n}$ passing through the origin. Let $\pi\,:\,\mathbb{C}^{n}-\{0\}\rightarrow\mathbb{P}(\mathbb{C}^{n}),\,\,\,z=(z_{1},...,z_{n})\mapsto[z]=[z_{1},...,z_{n}]=\mathbb{C}z$. We define a topology on $\mathbb{P}(\mathbb{C}^{n})$ by declaring $U\subseteq\mathbb{P}(\mathbb{C}^{n})$ to be open if and only if $\pi^{-1}(U)$ is open in $\mathbb{C}^{n}-\{0\}$. It can be shown that the family of maps \phi_{i}\,:\,\big{\{}[z_{1},...,z_{n}]\,\big{|}\,z_{i}\neq 0\big{\}}\rightarrow\mathbb{C}^{n-1},\,\,\,[z_{1},...,z_{n}]\mapsto\big{(}\tfrac{z_{1}}{z_{i}},...,\tfrac{z_{i-1}}{z_{i}},\tfrac{z_{i+1}}{z_{i}},...,\tfrac{z_{n}}{z_{i}}\big{)}, $i=1,...,n$, defines a complex atlas on $\mathbb{P}(\mathbb{C}^{n})$. The restriction of $\pi$ to the unit sphere $S^{2n-1}\subseteq\mathbb{R}^{2n}\cong\mathbb{C}^{n}$ yields a surjective submersion $\pi|_{S^{2n-1}}\,:\,S^{2n-1}\rightarrow\mathbb{P}(\mathbb{C}^{n})$ and hence there are tensors $g$ and $\omega$ on $\mathbb{P}(\mathbb{C}^{n})$ characterized by the formulas

[TABLE]

where $j:S^{2n-1}\hookrightarrow\mathbb{C}^{n}\cong\mathbb{R}^{2n}$ is the inclusion, $\textup{Re}\langle\cdot,\cdot\rangle$ and $\textup{Im}\langle\cdot,\cdot\rangle$ are the real and imaginary parts of the standard Hermitian product $\langle z,w\rangle=\overline{z}_{1}w_{1}+...+\overline{z}_{n}w_{n}$ on $\mathbb{C}^{n}$.

It can be shown that $(\mathbb{P}(\mathbb{C}^{n}),g,J)$ is a Kähler manifold, where $J$ is the associated complex structure, with fundamental form $\omega$.

3 Connections and connectors

This section follows closely [Dom62]. Let $M$ be a manifold.

Definition 3.1**.**

A linear connection $\nabla$ on $M$ is a map $\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathfrak{X}(M)$, $(X,Y)\mapsto\nabla_{X}Y$, satisfying the following properties:

1. (i)

$\nabla_{fX+gY}Z=f\nabla_{X}Z+g\nabla_{Y}Z$, 2. (ii)

$\nabla_{X}(Y+Z)=\nabla_{X}Y+\nabla_{X}Z$, 3. (iii)

$\nabla_{X}(fY)=X(f)Y+f\nabla_{X}Y$,

for all vector fields $X,Y,Z\in\mathfrak{X}(M)$ and for all functions $f,g\in C^{\infty}(M)$.

In local coordinates $(x_{1},...,x_{n})$ on $U\subseteq M$, if $X=\sum_{k=1}^{n}X^{i}\tfrac{\partial}{\partial x_{k}}$ and $Y=\sum_{k=1}^{n}Y^{k}\tfrac{\partial}{\partial x_{k}}$, then, by standard computations,

[TABLE]

where $\Gamma_{ij}^{k}:U\to\mathbb{R}$ are the Christoffel symbols, defined by the formula

[TABLE]

Let $\pi:TM\to M$ be the canonical projection and let $(U,\varphi)$ be a chart for $M$ with local coordinates $(x_{1},...,x_{n})$. Define $\widetilde{\varphi}:\pi^{-1}(U)\to\mathbb{R}^{2n}$ by

[TABLE]

Then $(\pi^{-1}(U),\widetilde{\varphi})$ is a chart for $TM$; let $(q_{1},...,q_{n},r_{1},...,r_{n})$ be the corresponding local coordinates (in particular, $q_{i}=x_{i}\circ\pi$ for every $i=1,...,n)$.

Let u=\sum_{k=1}^{n}u_{k}\tfrac{\partial}{\partial x_{k}}\big{|}_{p}\in\pi^{-1}(U) be arbitrary. Define a linear map $K_{u}:T_{u}(TM)\to T_{p}M$ by

[TABLE]

Lemma 3.2**.**

Let $X$ and $Y$ be vector fields on $M$. Suppose $Y(p)=u$. Then $K_{u}(Y_{*_{p}}X_{p})=(\nabla_{X}Y)(p)$.

Proof.

By standard computations,

[TABLE]

so

[TABLE]

where we have used $u_{j}=Y^{j}(p)$. Comparing the above formula with the local expression for $\nabla_{X}Y$ in coordinates (see (3.1)), one obtains the desired formula. ∎

Clearly, vectors of the form $Y_{*_{p}}X_{p}$, with $Y_{p}=u$, generate $T_{u}(TM)$, and so the above lemma implies that the definition of $K_{u}$ is independant of the choice of the chart $(U,\varphi)$.

The map

[TABLE]

defined for $A\in T_{u}(TM)$ by $K(A):=K_{u}(A)$, is called connector, or connection map, associated to $\nabla$.

The following result is an immediate consequence of the definition of $K$.

Proposition 3.3**.**

Let $K$ be the connector associated to a connection $\nabla$ on $M$. The following holds.

1. (i)

For every pair $X,Y$ of vector fields on $M$, $\nabla_{X}Y=KY_{*}X$, where $Y_{*}X$ denotes the derivative of $Y$ in the direction of $X$. 2. (ii)

For every $u\in T_{p}M$, the restriction of $K$ to $T_{u}(TM)$ is a linear map $T_{u}(TM)\to T_{p}M$.

If $A\in T_{u}(TM)$ is such that $\pi_{*_{u}}A=0$ and $K(A)=0$, then a simple calculation using local coordinates shows that $A=0$. Therefore,

Proposition 3.4**.**

Let $K$ be the connector associated to a connection $\nabla$ on $M$. Given $u\in T_{p}M$, the map $T_{u}(TM)\to T_{p}M\oplus T_{p}M$, defined by

[TABLE]

is a linear bijection.

Thus, given a linear connection $\nabla$, we can identify at any point $u\in T_{p}M$ the vector spaces $T_{u}(TM)$ and $T_{p}M\oplus T_{p}M$ via the map (3.2).

4 Dombrowski’s construction

Let $M$ be a smooth manifold endowed with a connection $\nabla$. We will denote by $\pi:TM\rightarrow M$ the canonical projection.

By Proposition 3.4, there is an identification of vector spaces $T_{u}(TM)\cong T_{p}M\oplus T_{p}M$, where $p=\pi(u)$. If there is no danger of confusion, we will therefore regard an element of $T_{u}(TM)$ as a pair $(v,w)$, where $v,w\in T_{p}M$.

Let $h$ be a Riemannian metric on $M$. The pair $(h,\nabla)$ determines an almost Hermitian structure on $TM$ via the following formulas:

[TABLE]

where $u,v,w,\overline{v},\overline{w}\;\in\;T_{p}M$.

The tensors $g,J,\omega$ are smooth (this will follow from their coordinate representation, see Proposition 4.5 below) and clearly, $J^{2}=-Id$, $g(Ju,Jv)=g(u,v)$ and $\omega(u,v)=g(Ju,v)$ for all $u,v\in TM$ such that $\pi(u)=\pi(v)$. Thus, $(TM,g,J)$ is an almost Hermitian manifold with fundamental form $\omega$. This is Dombrowski’s construction [Dom62].

We now review the analytical properties of Dombrowski’s construction. We begin with some definitions.

Definition 4.1**.**

A dualistic structure on a manifold $M$ is a triple $(h,\nabla,\nabla^{*})$, where $h$ is a Riemannian metric and where $\nabla$ and $\nabla^{*}$ are linear connections satisfying

[TABLE]

for all vector fields $X,Y,Z$ on $M$. The connection $\nabla^{*}$ is called the dual connection of $\nabla$ (and vice versa).

As the literature is not uniform, let us agree that the torsion $T$ and the curvature tensor $R$ of a connection $\nabla$ are defined as

[TABLE]

where $X,Y,Z$ are vector fields on $M$. By definition, a linear connection is flat if the torsion and curvature tensor are identically zero on $M$. A manifold endowed with a flat linear connection is called an affine manifold.

Definition 4.2**.**

A dualistic structure $(h,\nabla,\nabla^{*})$ is dually flat if both $\nabla$ and $\nabla^{*}$ are flat.

Proposition 4.3**.**

Let $(h,\nabla,\nabla^{*})$ be a dualistic structure on $M$ and let $(g,J,\omega)$ be the almost Hermitian structure on $TM$ associated to $(h,\nabla)$ via Dombrowski’s construction. The following are equivalent.

1. (i)

$(TM,g,J,\omega)$ is a Kähler manifold. 2. (ii)

$(M,h,\nabla,\nabla^{*})$ is dually flat.

Proof.

See [Dom62, Mol13]. ∎

We now direct our attention to the coordinate expressions for $g,J$ and $\omega$.

Definition 4.4**.**

Suppose $(M,\nabla)$ is an affine manifold. An affine coordinate system is a coordinate system $(x_{1},...,x_{n})$ defined on some open set $U\subseteq M$ such that

[TABLE]

for all $i,j=1,...,n$.

It can be shown that for every point $p$ in an affine manifold $M$, there is an affine coordinate system $(x_{1},...,x_{n})$ defined on some neighborhood $U\subseteq M$ of $p$ (see [Shi07]).

Proposition 4.5**.**

Let $(h,\nabla,\nabla^{*})$ be a dually flat structure on a manifold $M$ and let $(g,J,\omega)$ be the Kähler structure on $TM$ associated to $(h,\nabla)$ via Dombrowski’s construction. Let $x=(x_{1},...,x_{n})$ be an affine coordinate system with respect to $\nabla$ on $U\subseteq M$, and let $(q,r)=(q_{1},...,q_{n},r_{1},...,r_{n})$ denote the corresponding coordinates on $\pi^{-1}(U)$, as described before Lemma 3.2. Then, in the coordinates $(q,r)$,

[TABLE]

where h_{ij}=h\big{(}\tfrac{\partial}{\partial x_{i}},\tfrac{\partial}{\partial x_{j}}\big{)}, $i,j=1,...,n$.

Proof.

See [Mol14]. ∎

Corollary 4.6**.**

Under the hypotheses of Proposition 4.5, if $z_{k}:=q_{k}+ir_{k}$, $k=1,...,n$, then $(z_{1},...,z_{n})$ are complex coordinates on the complex manifold $(TM,J)$.

5 Ricci curvature

Let $N$ be a Kähler manifold with Kähler metric $g$. We denote by Ric the Ricci tensor of $g$,

[TABLE]

where $X,Y,Z$ are vector fields on $N$ and $R$ is the curvature tensor of $g$.

On the complexified tangent bundle $TN^{\mathbb{C}}=TN\otimes_{\mathbb{R}}\mathbb{C}$, we extend $\mathbb{C}$-linearly every tensor of $N$ at every point $p\;\in\;N$. For simplicity, we use the same symbols ($g$, Ric, etc) to indicate the corresponding $\mathbb{C}$-linear extensions.

Regarding local computations and indices, Greek indices $\alpha,\beta,\gamma$ shall run over $1,...,n$ while capital letters $A,B,C,...$ shall run over $1,...,n,\overline{1},\overline{2},...,\overline{n}$. Let $(z_{1},...,z_{n})$ be a system of complex coordinates on $N$. We denote by $x_{\alpha}$ and $y_{\alpha}$ the real and imaginary part of $z_{\alpha}$, i.e., $z_{\alpha}=x_{\alpha}+iy_{\alpha}$. With this notation, the vectors

[TABLE]

where $\alpha=1,...,n$, form a basis for $TN^{\mathbb{C}}$. Let $\textup{Ric}_{AB}=\textup{Ric}(Z_{A},Z_{B})$ be the components of the Ricci tensor in this basis. As it is well-known, these components are elegantly expressed via the following formulas:

[TABLE]

where $G:=(g_{\alpha\overline{\beta}})_{1\leq\alpha,\beta\leq n}$ is the associated Hermitian matrix.

We now specialize to the case $N=TM$, assuming that $g$ is the Kähler metric associated to a dually flat structure $(h,\nabla,\nabla^{*})$ on $M$ via Dombrowski’s construction.

Fix an affine coordinate system $(x_{1},...,x_{n})$ on an open set $U\subseteq M$ with respect to $\nabla$, and let $(q,r)=(q_{1},...,q_{n},r_{1},...,r_{n})$ be the corresponding coordinates on $\pi^{-1}(U)$, as described before Lemma 3.2, where $\pi:TM\to M$ is the canonical projection.

Given $1\leq\alpha\leq n$, define $z_{\alpha}:=q_{\alpha}+ir_{\alpha}$. Then $(z_{1},...,z_{n})$ are complex coordinates on $\pi^{-1}(U)\subseteq TM$. Applying (5.1), we obtain

[TABLE]

where $d$ is the determinant of the matrix $(h_{\alpha\beta})$. The second formula in (5.2) is the local expression for the Ricci tensor in the basis $\{Z_{\alpha},\overline{Z}_{\alpha}\}$. Returning to the coordinates $(q,r)$, a direct calculation using

[TABLE]

shows the following result (see [Mol14]).

Proposition 5.1**.**

Let $(h,\nabla,\nabla^{*})$ be a dually flat structure on $M$ and let $g$ be the Kähler metric on $TM$ associated to $(h,\nabla)$ via Dombrowski’s construction. If $x=(x_{1},...,x_{n})$ is an affine coordinate system on $M$ with respect to $\nabla$, then in the coordinates $(q,r)$, the matrix representation of the Ricci tensor of $g$ is

[TABLE]

and where $d$ is the determinant of the matrix h_{\alpha\beta}=h\big{(}\tfrac{\partial}{\partial x_{\alpha}},\tfrac{\partial}{\partial x_{\alpha}}\big{)}.

Recall that the scalar curvature is, by definition, the trace of the Ricci tensor.

Corollary 5.2**.**

Under the hypotheses of Proposition 5.1, the scalar curvature of $g$ is given in the coordinates $(q,r)$ by

[TABLE]

where $d$ is the determinant of the matrix $h_{\alpha\beta}$, and where $h^{\alpha\beta}$ are the coefficients of the inverse matrix of $h_{\alpha\beta}$.

Observe that the scalar curvature on $TM$ can be written $\textup{Scal}=S\circ\pi$, where $S:M\to\mathbb{R}$ is a globally defined function whose local expression is given by the right hand side of (5.4). The function $S$ is called Hessian scalar curvature (see [Shi07]).

6 Statistical manifolds

General references are [AJLS17, AN00, MR93].

Definition 6.1**.**

A statistical manifold is a pair $(S,j)$, where $S$ is a manifold and where $j$ is an injective map from $S$ to the space of all probability density functions $p$ defined on a fixed measure space $(\Omega,dx)$:

[TABLE]

If $\xi=(\xi_{1},...,\xi_{n})$ is a coordinate system on a statistical manifold $S$, then we shall indistinctly write $p(x;\xi)$ or $p_{\xi}(x)$ for the probability density function determined by $\xi$.

Given a “reasonable” statistical manifold $S$, it is possible to define a metric $h_{F}$ and a family of connections $\nabla^{(\alpha)}$ on $S$ $(\alpha\in\mathbb{R})$ in the following way: for a chart $\xi=(\xi_{1},...,\xi_{n})$ of $S$, define

[TABLE]

where $\mathbb{E}_{p_{\xi}}$ denotes the mean, or expectation, with respect to the probability $p_{\xi}dx$, and where $\partial_{i}$ is a shorthand for $\tfrac{\partial}{\partial\xi_{i}}$. It can be shown that if the above expressions are defined and smooth for every chart of $S$, then $h_{F}$ is a well defined metric on $S$ called the Fisher metric, and that the $\Gamma_{ij,k}^{(\alpha)}$’s define a connection $\nabla^{(\alpha)}$ via the formula $\Gamma_{ij,k}^{(\alpha)}(\xi)=(h_{F})_{\xi}(\nabla_{\partial_{i}}^{(\alpha)}\partial_{i},\partial_{j})$, which is called the $\alpha$-connection.

Among the $\alpha$-connections, the $(\pm 1)$-connections are particularly important; the 1-connection is usually referred to as the exponential connection, also denoted by $\nabla^{(e)}$, while the $(−1)$-connection is referred to as the mixture connection, denoted by $\nabla^{(m)}$.

In this paper, we will only consider statistical manifolds $S$ for which the Fisher metric and $\alpha$-connections are well defined.

Proposition 6.2**.**

Let $S$ be a statistical manifold. Then, $(h_{F},\nabla^{(\alpha)},\nabla^{(-\alpha)})$ is a dualistic structure on $S$. In particular, $\nabla^{(-\alpha)}$ is the dual connection of $\nabla^{(\alpha)}$.

Proof.

See [AN00]. ∎

We now recall the definition of an exponential family.

Definition 6.3**.**

An exponential family $\mathcal{E}$ on a measure space $(\Omega,dx)$ is a set of probability density functions $p(x;\theta)$ of the form

[TABLE]

where $C,F_{1}...,F_{n}$ are measurable functions on $\varOmega$, $\theta=(\theta_{1},...,\theta_{n})$ is a vector varying in an open subset $\Theta$ of $\mathbb{R}^{n}$ and where $\psi$ is a function defined on $\Theta$.

In the above definition it is assumed that the family of functions $\{1,F_{1},...,$ $F_{n}\}$ is linearly independent, so that the map $p(x,\theta)\mapsto\theta$ becomes a bijection, hence defining a global chart for $\mathcal{E}$. The parameters $\theta_{1},...,\theta_{n}$ are called the natural or canonical parameters of the exponential family $\mathcal{E}$.

Example 6.4** (Normal distribution).**

Normal distributions,

[TABLE]

form a $2$-dimensional statistical manifold, denoted by $\mathcal{N}$, parameterized by $(\mu,\sigma)\in\mathbb{R}\times\mathbb{R}_{+}^{*}$, where $\mu\in\mathbb{R}$ is the mean and $\sigma\in\mathbb{R}_{+}^{*}$ is the standard deviation (here \mathbb{R}_{+}^{*}:=\bigl{\{}x\in\mathbb{R}\bigl{|}x>0\bigr{\}}). It is an exponential family, because p(x;\mu,\sigma)=\exp\big{\{}\theta_{1}F_{1}(x)+\theta_{2}F_{2}(x)-\psi(\theta)\big{\}}, where

[TABLE]

Example 6.5**.**

Given a finite set $\Omega=\{x_{1},...,x_{n}\}$, define

[TABLE]

Elements of $\mathcal{P}_{n}^{\times}$ can be parametrized as follows: p(x;\theta)=\exp\big{\{}\sum_{i=1}^{n-1}\theta_{i}F_{i}(x)-\psi(\theta)\big{\}}, where

[TABLE]

Therefere $\mathcal{P}_{n}^{\times}$ is an exponential family of dimension $n-1$.

Example 6.6** (Binomial distribution).**

The set of binomial distributions defined over $\Omega:=\{0,...,n\}$,

[TABLE]

where $\binom{n}{k}=\tfrac{n!}{(n-k)!k!}$, is a $1$-dimensional statistical manifold, denoted by $\mathcal{B}(n)$, parametrized by q\in\bigl{(}0,1\bigr{)}. It is an exponential family, because p(k)=\exp\big{\{}C(k)+\theta F(k)-\psi(\theta)\big{\}}, where

[TABLE]

Proposition 6.7**.**

Let $\mathcal{E}$ be an exponential family such as in Definition 6.3. Then $(\mathcal{E},h_{F},\nabla^{(e)},\nabla^{(m)})$ is dually flat.

Proof.

See [AN00]. ∎

Corollary 6.8**.**

The tangent bundle $T\mathcal{E}$ of an exponential family $\mathcal{E}$ is a Kähler manifold for the Kähler structure $(g,J,\omega)$ associated to $(h_{F},\nabla^{(e)})$ via Dombrowski’s construction.

Proof.

Follows from Proposition 4.3. ∎

In the sequel, by the Kähler structure of $T\mathcal{E}$, we will implicitly refer to the Kähler structure of $T\mathcal{E}$ described in Corollary 6.8.

Example 6.9** ([Mol12, Mol13]).**

Let $\mathcal{P}_{n}^{\times}$ be the statistical manifold defined in Example 6.5. For an appropriate normalization of the Fubini-Study metric and symplectic form, it can be shown that there exists a map

[TABLE]

where $\mathbb{P}(\mathbb{C}^{n})^{\times}=\{[z_{1},...,z_{n}]\in\mathbb{P}(\mathbb{C}^{n})\,|\,z_{k}\neq 0\,\,\forall\,k=1,...,n\}$, with the following properties:

1. (i)

$\tau$ is a universal covering map whose Deck transformation group is isomorphic to $\mathbb{Z}^{n-1}$, 2. (ii)

$\tau$ is holomorphic and locally isometric.

In particular, if $\textup{Deck}(\tau)$ denotes the Deck transformation group of $\tau$, then $T\mathcal{P}_{n}^{\times}/\textup{Deck}(\tau)\cong\mathbb{P}(\mathbb{C}^{n})^{\times}$ (isomorphism of Kähler manifolds).

Example 6.10** (Binomial distribution [Mol13]).**

Let $\mathcal{B}(n)$ be the set of binomial distributions defined over $\Omega:=\{0,...,n\}$, as in Example 6.6. Let $S^{2}$ be the unit sphere in $\mathbb{R}^{3}$. Consider the map $\tau:T\mathcal{B}(n)\to(S^{2})^{\times}:=S^{2}-\{(\pm 1,0,0)\}$ given by

[TABLE]

where $(q,r)$ are the coordinates on $T\mathcal{B}(n)$ associated to the natural parameter $\theta$, as described before Lemma 3.2.

It is easy to check that if the Kähler structure of $S^{2}$ (as described in Example 2.7) is multiplied by $n$, then $\tau$ is a holomorphic and locally isometric universal covering map whose Deck transformation group is $\textup{Deck}(\tau)\cong\mathbb{Z}$. Therefore $T\mathcal{B}(n)/\textup{Deck}(\tau)\cong(S^{2})^{\times}$ (isomorphism of Kähler manifolds).

Example 6.11** (Normal distributions [Mol14]).**

Let $\mathcal{N}$ be the set of Gaussian distributions, as defined in Example 6.4. As a complex manifold, $T\mathcal{N}$ is the product $\mathbb{H}\times\mathbb{C}$, where $\mathbb{H}:=\{\tau\in\mathbb{C}\>|\>Im(\tau)>0\}$ is the Poincaré upper half-plane. The metric of the space $\mathbb{H}\times\mathbb{C}$ is the Kähler-Berndt metric $g_{KB}$, which can be described as follows. If $\tau=u+iv\in\mathbb{H}$ and $z=x+iy\in\mathbb{C}$, then in the coordinates \bigl{(}u,v,x,y\bigr{)},

[TABLE]

This metric plays an important role in the context of Number Theory, in relation to the so-called Jacobi forms [BS98, EZ85].

We end this section with some technical results that we will use in the next section.

Let $\mathcal{E}$ be an exponential family of dimension $n$ defined over the measure space $(\Omega,dx)$, with elements of the form

[TABLE]

where $C,F_{1},...,F_{n}$ are measurable functions on $\varOmega$, $\theta=(\theta_{1},...,\theta_{n})$ is a vector in an open subset $\Theta$ of $\mathbb{R}^{n}$ and where $\psi$ is a function defined on $\Theta$.

Given $i=1,...,n$, we defined $\eta_{i}:\mathcal{E}\to\mathbb{R}$ by

[TABLE]

The functions $\eta_{1},...,\eta_{n}$ are called expectation parameters. Note that, if the functions $F_{i}:\Omega\to\mathbb{R}$ are not measurable, the existence of the functions $\eta_{i}$ is not guaranteed. However, in the particular case where $\Omega$ is finite, the functions $\eta_{i}$ exist and have good properties, as described in the following result.

Proposition 6.12**.**

Let $\mathcal{E}$ be an exponential family defined over a finite set $\Omega=\{x_{0},x_{1},...,x_{m}\}$ endowed with the counting measure, with $C,F_{1},...,F_{n}:\Omega\to\mathbb{R}$, $\theta\>\in\>\Theta$ and $\psi:\Theta\to\mathbb{R}$ as above. The following holds.

1. (i)

The set $\Theta$ can be taken equal to $\mathbb{R}^{n}$. 2. (ii)

\bigl{(}\eta_{1},...,\eta_{n}\bigr{)} is a global system of affine coordinates with respect to $\nabla^{(m)}$. 3. (iii)

$\dfrac{\partial\psi}{\partial\theta_{i}}=\eta_{i}$        for all $i=1,...,n$, 4. (iv)

\dfrac{\partial^{2}\psi}{\partial\theta_{i}\partial\theta_{j}}=\dfrac{\partial\eta_{i}}{\partial\theta_{j}}=h_{F}\Big{(}\dfrac{\partial}{\partial\theta_{i}},\dfrac{\partial}{\partial\theta_{j}}\Big{)},        for all $i,j=1,...,n$. 5. (v)

\psi(\theta)=\ln\biggl{\{}\displaystyle\sum_{k=0}^{m}\exp\biggl{(}C\bigl{(}x_{k}\bigr{)}+\sum_{j=1}^{n}\theta_{j}F_{j}\bigl{(}x_{k}\bigr{)}\biggr{)}\biggr{\}}         for all $\theta\in\mathbb{R}^{n}$.

Proof.

See [AN00]. ∎

7 Constant scalar curvature

Let $\Omega=\{x_{0},x_{1},...,x_{m}\}$ be a finite set endowed with the counting measure $dx$. Let $\mathcal{E}$ be a 1-dimensional exponential family defined over $(\Omega,dx)$, with elements of the form

[TABLE]

where $C,F:\Omega\to\mathbb{R}$ are functions, $\theta\in\mathbb{R}$ and $\psi:\mathbb{R}\to\mathbb{R}$ is a function. We denote by $\eta:\mathcal{E}\to\mathbb{R}$ the expectation parameter.

We will use the following notations:

• •

$F_{i}:=F(x_{i})$, $C_{i}:=C(x_{i})$, $i=0,...,m$,

• •

$F_{min}:=min\{F_{0},...,F_{m}\}$,  $F_{max}:=max\{F_{0},...,F_{m}\}$,

• •

$I:=\{0,1,...,m\}$,

• •

$I_{min}:=\{k\in I\>|\>F_{k}=F_{min}\}$,

• •

$I_{max}:=\{k\in I\>|\>F_{k}=F_{max}\}$.

Note that $F_{min}\neq F_{max}$ (since the functions $1$ and $F$ are assumed to be linearly independent). Note also that $I_{min}\neq\emptyset$ and $I_{max}\neq\emptyset$.

Lemma 7.1**.**

We have

[TABLE]

In particular, $\eta$ is a bounded function.

Proof.

By Proposition 6.12, (iii) and (iv), we have

[TABLE]

Multiplying the numerator and denominator by $e^{-\theta F_{min}}$ yields

[TABLE]

where $I_{min}^{c}=I-I_{min}$. If $k\in I_{min}^{c}$, then $F_{k}-F_{min}>0$ and so,

[TABLE]

Thus,

[TABLE]

Analogously,

[TABLE]

The lemma follows. ∎

Let $(g,J,\omega)$ be the Kähler structure on $T\mathcal{E}$ associated to \bigl{(}h_{F},\nabla^{(e)}\bigr{)} via Dombrowski’s construction. We denote by $\textup{Scal}:T\mathcal{E}\to\mathbb{R}$ the corresponding scalar curvature.

Proposition 7.2**.**

Suppose the scalar curvature of $T\mathcal{E}$ is constant and equal to $\lambda\in\mathbb{R}$. Then $\lambda\neq 0$ and there exist $a,b,r,s\in\mathbb{R}$, with $a\neq b$, such that

[TABLE]

for all $\theta\in\mathbb{R}$. Consequently, the coordinate expression for the Fisher metric with respect to $\theta$ is

[TABLE]

where $\textup{cosh}(x)=\tfrac{e^{x}+e^{-x}}{2}$ is the hyperbolic cosine function.

Proof.

By Corollary 5.2,

[TABLE]

where h_{F}(\theta):=h_{F}\bigl{(}\frac{\partial}{\partial\theta},\frac{\partial}{\partial\theta}\bigr{)}, and so,

[TABLE]

where we have used $h_{F}=\tfrac{\partial\eta}{\partial\theta}$ (see Proposition 6.12). Integrating we obtain

[TABLE]

which is equivalent to

[TABLE]

where $c_{2},c_{3}\in\mathbb{R}$. We conclude that $\textup{Scal}\equiv\lambda$ if and only if there exist $a,b\in\mathbb{R}$ such that

[TABLE]

Because {\displaystyle\tfrac{\partial\eta}{\partial\theta}=h_{F}\bigl{(}\theta\bigr{)}>0}, Equation (7.2) implies that

[TABLE]

Therefore we can divide both sides of (7.2) by (7.3). This yields

[TABLE]

Hence it all boils down to integrate the function

[TABLE]

Let $\triangle=a^{2}-4b(-\tfrac{1}{2}\lambda)=a^{2}+2b\lambda$ be the discriminant of the polynomial $-\tfrac{1}{2}\lambda x^{2}+ax+b$. We will consider 3 cases.

Case 1: $\triangle<0$. Integration of (7.4) yields:

[TABLE]

The left hand side of this equation is a bounded function, whereas the right hand side is not. Thus this case is not possible.

Case 2: $\triangle=0$. First, suppose that $\lambda=0$. Then the condition $\triangle=0$ implies that $a=0$, which also implies by (7.3) that $b>0$. On the other hand, it follows from (7.2) that \eta\bigl{(}\theta\bigr{)}=b\theta+c. By Lemma 7.1, the function $\eta$ is bounded, whereas the function $b\theta+c$ is not (since $b\neq 0$). It follows that $\lambda=0$ is impossible.

Now suppose that $\lambda\neq 0$. In this case, there exists $\alpha\in\mathbb{R}$ such that

[TABLE]

Integrating (7.4) we obtain

[TABLE]

Putting $\theta=-c$ in (7.5) yields $\tfrac{1}{\eta(-c)+\alpha}=0$, which is not possible.

Case 3: $\triangle>0$. Suppose first that $\lambda=0$ (in particular, this implies $a\neq 0$). Then, by (7.4),

[TABLE]

and so, there exists $c\in\mathbb{R}-\{0\}$ such that

[TABLE]

which is not possible, since $\eta$ is bounded.

Suppose that $\lambda\neq 0$. In this case, there exist $\alpha,\beta\in\mathbb{R}$ with $\alpha<\beta$, such that

[TABLE]

Then, integration of (7.4) yields

[TABLE]

and so

[TABLE]

It follows from (7.6) that \alpha,\beta\notin\textup{Im}\bigl{(}\eta\bigr{)}. Therefore we have the following possibilities:

(i) \textup{Im}\bigl{(}\eta\bigr{)}\subseteq\bigl{(}-\infty,\alpha\bigr{)}\cup\bigl{(}\beta,+\infty\bigr{)}.

In this case, (7.6) becomes

[TABLE]

Putting $\theta=-c$ we obtain $\alpha=\beta$ which is a contradiction.

(ii) $\textup{Im}(\eta)\subseteq(\alpha,\beta)$. In this case,

[TABLE]

from which it follows that

[TABLE]

Integrating again (remember that $\frac{\partial\psi}{\partial\theta}=\eta$) we obtain

[TABLE]

where \tfrac{2}{\lambda}\ln\bigl{(}e^{\omega}\bigr{)} is a constant. Then,

[TABLE]

Letting $a:=\frac{\lambda}{2}\alpha$, $b:=\frac{\lambda}{2}\beta$, $r:=\frac{\lambda}{2}\alpha c+\omega$ and $s:=\frac{\lambda}{2}\beta c+\omega$, yields the desired result. ∎

Corollary 7.3**.**

If the scalar curvature of $T\mathcal{E}$ is constant and equal to $\lambda\in\mathbb{R}$, then $\lambda>0$.

Proof.

This follows immediately from the formula h_{F}(\theta)=\tfrac{(a-b)^{2}}{2\lambda\,\textup{cosh}^{2}\big{(}\tfrac{a-b}{2}\theta+\tfrac{r-s}{2}\big{)}} and the fact that $h_{F}(\theta)>0$ for all $\theta\in\mathbb{R}$ (since $h_{F}$ is a metric). ∎

Remark 7.4**.**

If \psi(\theta)=\tfrac{2}{\lambda}\ln\bigl{\{}e^{a\theta+r}+e^{b\theta+s}\bigr{\}}, with $a<b$, then $a=\tfrac{\lambda}{2}F_{min}$ and $b=\tfrac{\lambda}{2}F_{max}$. To see this, it suffices to compute $\textup{lim}_{\theta\to\pm\infty}\,\eta(\theta)=\textup{lim}_{\theta\to\pm\infty}\tfrac{\partial\psi}{\partial\theta}$ and to compare with Lemma 7.1.

Lemma 7.5**.**

Let $\alpha$ be a nonzero real number and $f:\mathbb{R}\to\mathbb{R},$ $x\mapsto(1+e^{x})^{\alpha}$. Given $k\in\mathbb{N}=\{0,1,...\}$, let $\mathcal{A}_{k}$ be the linear subspace of $C^{\infty}(\mathbb{R})$ spanned by $f$, $f^{\prime}$,…,$f^{(k)}$ (derivatives of $f$), that is,

[TABLE]

If $\alpha\notin\mathbb{N}$, then $\textup{dim}\mathcal{A}_{k}\geq k+1$.

Proof.

It is easy to see that if $\alpha\not\in\mathbb{N}$, then the family of linearly independent functions

[TABLE]

is contained in $\mathcal{A}_{k}$. ∎

Proposition 7.6**.**

Suppose that the scalar curvature of $T\mathcal{E}$ is constant and equal to $\lambda\in\mathbb{R}$. Then there exists a positive integer $d$ such that $\lambda=\tfrac{2}{d}.$

Proof.

From Proposition 6.12, there exist $a,b,r,s\in\mathbb{R}$ such that

[TABLE]

for all $\theta\in\mathbb{R}$. On the other hand, it follows from Proposition 6.12 that

[TABLE]

Thus

[TABLE]

that is,

[TABLE]

Multiplying by $e^{-\frac{2}{\lambda}(a\theta+r)}$ we obtain

[TABLE]

where $\alpha:=\frac{2}{\lambda}>0$, $x:=(b-a)\theta+s-r$, $\xi_{k}:=F_{k}-\frac{2}{\lambda}a$ and $\omega_{k}:=C_{k}-\frac{2}{\lambda}r$. Since $\theta=\tfrac{x-(s-r)}{b-a}$, this can be rewritten as

[TABLE]

where $\xi_{k}^{{}^{\prime}}:=\frac{\xi_{k}}{b-a}$ and $\omega_{k}^{{}^{\prime}}:=-\frac{s-r}{b-a}\xi_{k}+\omega_{k}$. Note that (7.7) holds for all $x\in\mathbb{R}$.

Consider the linear subspace of $C^{\infty}(\mathbb{R})$ spanned by the functions $e^{\xi_{k}^{{}^{\prime}}x+\omega_{k}^{{}^{\prime}}}$, $k=0,...,m$, that is,

[TABLE]

Observe that $\textup{dim}E\leq m+1$ and that for every $h\in E$, the derivative of $h$ with respect to $x$ belongs to $E$, that is, $\frac{dh}{dx}\in E$.

Let $f:\mathbb{R}\to\mathbb{R}$ be the function defined by $f(x):=(1+e^{x})^{\alpha}$. Because of (7.7), $f$ belongs to $E$, and by the observation above, so does its derivatives of all orders. Therefore $\mathcal{A}_{k}:=\textup{Span}\{f,f^{\prime},f^{\prime\prime},...,f^{(k)}\}$ is a linear subspace of $E$ for every integer $k\geq 0$, which implies $\textup{dim}\,\mathcal{A}_{k}\leq m+1$ for every $k$. According to Lemma 7.5, this is only possible if $\alpha\in\mathbb{N}$, that is, if $\frac{2}{\lambda}\in\mathbb{N}$. ∎

In what follows, we will use the following notations:

• •

$\alpha_{0},...,\alpha_{p}$ are the unique real numbers such that $\alpha_{0}<...<\alpha_{p}$ and $\textup{Im}(F)=\{\alpha_{0},...,\alpha_{p}\},$

• •

$I_{i}:=\{k\;\in\;I\>|\>F_{k}=\alpha_{i}\}$, $i=0,...,p$,

• •

$e^{\omega_{i}}=\underset{k\;\in\;I_{i}}{\sum}e^{C_{k}}$.

Note that $p\leq m$ and that $\alpha_{0}=F_{min}$ and $\alpha_{p}=F_{max}$.

Lemma 7.7**.**

Assume that $\textup{Scal}\equiv\frac{2}{d}$ on $T\mathcal{E}$, with $d\in\mathbb{N}^{*}$. With the notation of Proposition 7.6, we have

[TABLE]

for every $\theta\in\mathbb{R}$.

Proof.

We know from Proposition 7.2 and Proposition 7.6 that there are real numbers $a,b,r,s$, with $a<b$, and $d\in\mathbb{N}^{*}$ such that

[TABLE]

for all $\theta\in\mathbb{R}$, where we have used $a=\tfrac{\lambda}{2}F_{min}=\tfrac{\alpha_{0}}{d}$ and $b=\tfrac{\lambda}{2}F_{max}=\tfrac{\alpha_{p}}{d}$ (see Remark 7.4). From Proposition 6.12, we also have that \psi(\theta)=\ln\big{(}\sum_{k=0}^{m}e^{C_{k}+\theta F_{k}}\big{)}. Therefore

[TABLE]

We compute the left and right hand sides of (7.8) separately:

[TABLE]

where we have used $e^{\omega_{i}}=\underset{k\;\in\;I_{i}}{\sum}e^{C_{k}}$. The lemma follows by comparing the left and right hand sides. ∎

Lemma 7.8**.**

Let $\{\xi_{i},\eta_{i}\,\,|\,\,i=0,...,d\}$ and $\{\alpha_{j},\omega_{j}\,\,|\,\,j=0,...,p\}$ be families of real numbers such that $\xi_{0}<\cdots<\xi_{d}$ and $\alpha_{0}<\cdots<\alpha_{p}$. If

[TABLE]

for all $\theta\in\mathbb{R}$, then $d=p$ and for every $i=0,...,d$, $\xi_{i}=\alpha_{i}$ and $\eta_{i}=\omega_{i}$.

Proof.

By hypothesis, we have

[TABLE]

for all $\theta\in\mathbb{R}$, where $F,G:\mathbb{R}\to\mathbb{R}$ are functions that are easily seen to satisfy $\textup{lim}_{\theta\to\infty}F(\theta)=1$ and $\textup{lim}_{\theta\to\infty}G(\theta)=1$. It follows that

[TABLE]

which forces $\xi_{d}=\alpha_{p}$ and $\eta_{d}=\omega_{p}$. The lemma is proved by repeating the same argument. ∎

Theorem 7.9**.**

Let $\mathcal{E}$ be a 1-dimensional exponential family defined over a finite set $\Omega=\{x_{0},...,x_{m}\}$, with elements of the form $p(x;\theta)=\textup{exp}\{C(x)+\theta F(x)-\psi(\theta)\}$, where $C,F:\Omega\to\mathbb{R}$, $\theta\in\mathbb{R}$ and $\psi:\mathbb{R}\to\mathbb{R}$. Suppose that $\textup{Im}(F)=\{\alpha_{0}<...<\alpha_{p}\}$. Given $i=0,...,p$, define $\omega_{i}\in\mathbb{R}$ via the formula

[TABLE]

Then the scalar curvature $\textup{Scal}:T\mathcal{E}\to\mathbb{R}$ is constant if and only if there exist $r,s\in\mathbb{R}$ such that

[TABLE]

for all $k=0,...,p$, and in that case, $\textup{Scal}\equiv\frac{2}{p}$.

Proof.

$(\Rightarrow)$ This follows from Lemma 7.7 and Lemma 7.8. $(\Leftarrow)$ This can be proved by reversing the reasoning above. ∎

Corollary 7.10**.**

Let $\mathcal{E}$ be a 1-dimensional exponential family defined over a finite set \Omega=\bigl{\{}x_{0},...,x_{m}\bigr{\}}. If the scalar curvature of $T\mathcal{E}$ is constant and equal to $\lambda$, then

[TABLE]

Example 7.11**.**

(Binomial distribution). Recall that elements of $\mathcal{B}(n)$ are parametrized as follows

[TABLE]

where k\in\bigl{\{}0,...,n\bigr{\}} and $\theta=\ln(\tfrac{q}{1-q})\in\mathbb{R}$. In this case, we have $p=n$ and

[TABLE]

for all $k=0,...,n$. Clearly $\alpha_{0},\alpha_{1},...,\alpha_{n}$ and $\omega_{0},...,\omega_{n}$ are solutions of (7.11) with $r=s=0$. Therefore the scalar curvature of $T\mathcal{B}(n)$ is constant and equal to $\frac{2}{n}$.

8 Equivalent and reduced exponential families

The following notation will be used throughout this section. Given a finite set $\Omega=\{x_{0},...,x_{m}\}$, let $C(\Omega)$ denote the space of maps $\Omega\to\mathbb{R}$ (clearly there is a natural identification $C(\Omega)\cong\mathbb{R}^{m+1}$). Given $F,C\in C(\Omega)$, let $\mathcal{E}_{C,F}$ denote the 1-dimensional exponential family defined over $\Omega$ with elements of the form

[TABLE]

where $x\in\Omega$, $\theta\in\mathbb{R}$ and \psi_{C,F}(\theta)=\ln\big{(}\sum_{k=0}^{m}\textup{exp}(C(x_{k})+\theta F(x_{k}))\big{)}. In the above notation, it is assumed that the function $F$ and the constant function $\mathds{1}:\Omega\to\mathbb{R},$ $x\mapsto 1$ are linearly independent (this guarantees that the map $\mathbb{R}\to\mathcal{E}_{C,F}$, $\theta\mapsto p_{C,F}(\,.\,;\theta)$ is bijective). In other words, $F\in\mathbb{R}^{m+1}-\mathbb{R}\cdot\mathds{1}$.

Definition 8.1**.**

Two 1-dimensional exponential families $\mathcal{E}_{C,F}$ and $\mathcal{E}_{C^{\prime},F^{\prime}}$ defined over the same set $\Omega=\{x_{0},...,x_{m}\}$ are equivalent if the families of maps $\{p_{C,F}(\,.\,;\theta)\,:\,\Omega\to\mathbb{R}\}_{\theta\in\mathbb{R}}$ and $\{p_{C^{\prime},F^{\prime}}(\,.\,;\theta)\,:\,\Omega\to\mathbb{R}\}_{\theta\in\mathbb{R}}$ coincide.

In order to caracterize equivalent exponential families, we introduce the group of matrices

[TABLE]

Given an integer $m\geq 1$, the group $G$ acts on $U_{m}:=\mathbb{R}^{m+1}\times(\mathbb{R}^{m+1}-\mathbb{R}\cdot\mathds{1})$ via the formula

[TABLE]

where $C\in\mathbb{R}^{m+1}$ and $F\in\mathbb{R}^{m+1}-\mathbb{R}\cdot\mathds{1}$.

Proposition 8.2**.**

Two 1-dimensional exponential families $\mathcal{E}_{C,F}$ and $\mathcal{E}_{C^{\prime},F^{\prime}}$ defined over the same finite set $\Omega$ are equivalent if and only if there exists g=\Big{[}\begin{smallmatrix}1&b&d\\ 0&a&c\\ 0&0&1\end{smallmatrix}\Big{]}\in G such that $(C,F)=g\cdot(C^{\prime},F^{\prime})$. In that case, the following holds.

1. (i)

$p_{C,F}(x;\theta)=p_{C^{\prime},F^{\prime}}(x;a\theta+b)$      for all $x\in\Omega$ and all $\theta\in\mathbb{R}$. 2. (ii)

$\psi_{C,F}(\theta)=\psi_{C^{\prime},F^{\prime}}(a\theta+b)+c\theta+d$      for all $\theta\in\mathbb{R}$.

Proof.

$(\Rightarrow)$ Suppose $\mathcal{E}_{C,F}\sim\mathcal{E}_{C^{\prime},F^{\prime}}$. Because the maps $\mathbb{R}\to\mathcal{E}_{C,F}$, $\theta\mapsto p_{C,F}(\,.\,;\theta)$ and $\mathbb{R}\to\mathcal{E}_{C^{\prime},F^{\prime}}$, $\theta^{\prime}\mapsto p_{C^{\prime},F^{\prime}}(\,.\,;\theta^{\prime})$ are bijective, there exists a bijection $\phi:\mathbb{R}\to\mathbb{R}$ such that

[TABLE]

for all $\theta\in\mathbb{R}$ and all $x\in\Omega.$ Since $F^{\prime}$ and $\mathds{1}$ are linearly independent, there exist $y,z\in\Omega$ such that $F^{\prime}(y)\neq F^{\prime}(z)$. Putting $x=y$ and $x=z$ in (8.1), we obtain the following system

[TABLE]

Subtracting, we obtain

[TABLE]

for all $\theta\in\mathbb{R}$. Therefore there exist $a,b\in\mathbb{R}$ such that $\phi(\theta)=a\theta+b$ for all $\theta\in\mathbb{R}$. Note that $a$ is necessarily nonzero. Taking the derivative in (8.1) with respect to $\theta$ and using the formula $\phi(\theta)=a\theta+b$ we find that

[TABLE]

for all $\theta\in\mathbb{R}$ and all $x\in\Omega.$ This implies that there exists $c\in\mathbb{R}$ such that

[TABLE]

for all $x\in\Omega$, and

[TABLE]

for all $\theta\in\mathbb{R}$. Integrating the equation above, we obtain

[TABLE]

for all $\theta\in\mathbb{R}$, where $d\in\mathbb{R}$ is some constant. Then, using (8.1), (8.3) and (8.4) we see that

[TABLE]

for all $x\in\Omega$. It follows from (8.3) and (8.5) that (C,F)=\Big{[}\begin{smallmatrix}1&b&d\\ 0&a&c\\ 0&0&1\end{smallmatrix}\Big{]}\cdot(C^{\prime},F^{\prime}), which concludes one direction of the proof. Note that the computation above shows that if $\mathcal{E}_{C,F}\sim\mathcal{E}_{C^{\prime},F^{\prime}}$, then (i) and (ii) hold.

$(\Leftarrow)$ Left as a simple exercice to the reader. ∎

Definition 8.3**.**

Let $V$ be a finite dimensional real vector space and let $k$ be an integer satisfying $1\leq k\leq\textup{dim}\,V$. The affine Grassmannian, denoted by $\textup{Graff}_{k}(V)$, is the set of all $k$-dimensional affine subspaces of $V$.

It can be shown that $\textup{Graff}_{k}(V)$ is a noncompact smooth manifold of dimension $(n-k)(k+1)$, where $n=\textup{dim}\,V$ (see [LWY19]).

Given an integer $m\geq 1$, the space $\mathbb{R}^{m+1}$ decomposes as the following direct sum:

[TABLE]

where $V_{m}$ is the orthogonal complement of $\mathds{1}=(1,...,1)$ in $\mathbb{R}^{m+1}$ with respect to the usual inner product $\langle,\rangle$ on $\mathbb{R}^{m+1}$, that is,

[TABLE]

Given $u\in\mathbb{R}^{m+1}$, we will denote by $u^{\perp}\in V_{m}$ the orthogonal projection of $u$ on $V_{m}$.

Finally, given $(C,F)\in U_{m}$, we will denote by $[C,F]$ the corresponding equivalence class in the quotient space $U_{m}/G$.

Proposition 8.4**.**

For every integer $m\geq 1$, the map

[TABLE]

given by $f([C,F]):=C^{\perp}+\textup{span}\{F^{\perp}\}$ is a bijection.

Proof.

By a direct verification. ∎

It follows from Proposition 8.2 and Proposition 8.4 that the set of equivalence classes of 1-dimensional exponential families defined over the same finite set $\Omega=\{x_{0},...,x_{m}\}$ is in one-to-one correspondence with $\textup{Graff}_{1}(V_{m})$.

Example 8.5**.**

All 1-dimensional exponential families defined over $\Omega=\{x_{0},x_{1}\}$ are equivalent, because $\textup{Graff}_{1}(V_{1})=\{V_{1}\}$ is a single point.

Definition 8.6**.**

Let $\mathcal{E}=\mathcal{E}_{C,F}$ be a 1-dimensional exponential family defined over a finite set $\Omega=\{x_{0},...,x_{m}\}$. Let $\{\alpha_{i}\}_{i=0,...,p}$ and $\{\omega_{i}\}_{i=0,...,p}$ be the families of real numbers characterized by the following conditions:

1. (i)

$\textup{Im}\,F=\{\alpha_{0},...,\alpha_{p}\}$ and $\alpha_{0}<...<\alpha_{p}$, 2. (ii)

$e^{\omega_{i}}=\sum_{k\in F^{-1}(\alpha_{i})}e^{C(x_{k})}$.

Let $\Omega_{red}=\{0,1,...,p\}$. Define $C_{red},F_{red}:\Omega_{red}\to\mathbb{R}$ by

[TABLE]

where $k=0,...,p$. Then $\mathcal{E}_{red}:=\mathcal{E}_{C_{red},F_{red}}$ is a 1-dimensional exponential family defined over $\Omega_{red}$. We call it the reduced exponential family of $\mathcal{E}$.

Remark 8.7**.**

If $\mathcal{E}=\mathcal{E}_{C,F}$ is a 1-dimensional exponential family defined over a finite set $\Omega$, then $\psi_{C,F}(\theta)=\psi_{C_{red},F_{red}}(\theta)$ for all $\theta\in\mathbb{R}$.

Proposition 8.8**.**

Let $\mathcal{E}=\mathcal{E}_{C,F}$ be a 1-dimensional exponential family defined over a finite set $\Omega=\{x_{0},...,x_{m}\}$. The following are equivalent.

1. (i)

The scalar curvature of $T\mathcal{E}$ is constant. 2. (ii)

$\mathcal{E}_{red}\sim\mathcal{B}(p)$, where $p+1$ is the cardinality of $\Omega_{red}$.

Proof.

Let $\Omega_{B}=\{0,1,...,p\}$ and let $C_{B},F_{B}\,:\,\Omega_{B}\to\mathbb{R}$ be defined by $C_{B}(k)=\ln\binom{p}{k}$ and $F_{B}(k)=k.$ Comparing with Example 6.6, we see that $\mathcal{E}_{C_{B},F_{B}}\sim\mathcal{B}(p)$.

If the scalar curvature of $T\mathcal{E}$ is constant, then by Theorem 7.9 there are real numbers $r$ and $s$ such that

[TABLE]

for all $k=0,...,p$, which implies that

[TABLE]

It follows from this and Proposition 8.2 that $\mathcal{E}_{red}\sim\mathcal{E}_{C_{B},F_{B}}\sim\mathcal{B}(p)$.

Conversely, if $\mathcal{E}_{red}\sim\mathcal{B}(p)$, then $\mathcal{E}_{red}\sim\mathcal{E}_{C_{B},F_{B}}$ and hence there is g=\Big{[}\begin{smallmatrix}1&b&d\\ 0&a&c\\ 0&0&1\end{smallmatrix}\Big{]} such that $(C_{red},F_{red})=g\cdot(C_{B},F_{B})$, which implies that $C_{red}(k)$ and $F_{red}(k)$ are solutions of (8.8) for all $k=0,...,p$, provided $r=b+\tfrac{d}{p}$ and $s=\tfrac{d}{p}$. By Theorem 7.9 again, this implies that the scalar curvature of $T\mathcal{E}$ is constant. ∎

Remark 8.9**.**

As we saw in this paper, if the scalar curvature $\textup{Scal}:T\mathcal{E}\to\mathbb{R}$ of the tangent bundle of a 1-dimensional exponential family defined over a finite set is constant, then $\textup{Scal}>0$. This is not true for more general exponential families. For example, if $\mathcal{E}=\mathcal{N}$ is the family of Gaussian distributions over $\mathbb{R}$ (see Example 6.4), then $\textup{Scal}:T\mathcal{N}\to\mathbb{R}$ is constant and equal to $-6$ (see [Mol14]).

Acknowledgments

I am thankful to Caroline Santos Leite Ribeiro who carefully read and helped typing a preliminary version of this article.