数理解析研究所講究録
第 822 巻 1993 年 32-42

32

Dynamical Systems on Statistical Models
Akio Fujiwara (藤原彰夫)
Department of Mathematical Engineering and Information Physics

University of Tokyo, Tokyo 113, Japan
Abstract

Dualistic properties of a gradient flow on a manifold $M$ associated with a dualistic
structure$(g, \nabla, \nabla^{*})$is studied from an information geometrical viewpoint. Statisticalsignificance of the gradient flow is also investigated.

1

Introduction

Motivated mainly by classical mechanics, completely integrable dynamical systems havebeen investigated by many researchers. Furthermore, some authors have sought contacts
with other fields such as linear programming [4] and eigenvalue problems of matrices [5],see also [6] and the references cited therein.
On the other hand, some physicists have studied nonequilibrium or dissipative processes
from a geometrical viewpoint [3]. Obata et al. also examined some nonequilibrium pro-

cesses by using information geometry [9]. They showed the Uhlembeck-Ornstein process
is a geodesic motion with respect to the exponential connection on a Gaussian model.
Quite recently, Nakamura pointed out that certain gradient flows on Gaussian andmultinomial distributions can be characterized as completely integrable Hamiltonian systems [8]. This is the first suggestion of the connection between two seemingly unrelated

fields, i.e., information geometry and completely integrable dynamical systems.

In this paper, general dualistic properties of a gradient flow on a manifold $M$ associatedwith a dualistic structure$(g, \nabla, \nabla^{*})$is studied from an information geometrical point ofview. Statistical significance of the gradient flow is also investigated.

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2

Dualistic geometry

We first give a brief summary of dualistic geometry. For details, consult [2]. Let $M$ be a
Riemannian manifold with metric . Two affine connections$\nabla$$g$be dual with respect to$g$

if for any vector field $A,$ $B$ , and$C$

on $M$ ,$Ag(B|C)=g(\nabla_{A}B|C)+g(B|\nabla_{A}^{*}C)$

where $g(B|C)$ denotes the inner product of

$B$

and

$C$

and V’ on $M$ are said to,with respect to the metric . If the$g$torsions and the Riemannian curvatures of $M$ with respect to the connections$\nabla$and V’

vanish, $M$ is said to be flat, and a pair of divergences on $M$ are defined in the following

way. We first construct mutually dual affine coordinates on $M$ , i.e.,$\theta=[\theta^{i}]$and V’-affine coordinate$\eta=[\eta_{i}]$$M$$\partial_{i}=\partial/\partial\theta^{i}$and$\partial^{j}=\partial/\partial\eta_{j}$-affine coordinate

which satisfy$g(\partial_{i}|\partial^{j})=\delta_{i}^{j}$where$\nabla$,(1). Then there exist such potential functions$\psi(\theta),$$\phi(\eta)$onsatisfying$\theta^{i}=\partial_{i}\phi(\eta)$

where$\theta\cdot\eta=\theta^{i}\eta_{i}$,$\eta_{i}=\partial^{i}\psi(\eta)$,$\psi(\theta)+\phi(\eta)-\theta\cdot\eta=0$. By using these potentials, we define the V-divergence$D(p_{1}||p_{2})=\psi(\theta_{2})+\phi(\eta_{1})-\theta_{2}\cdot\eta_{1}$where$\eta_{1}$and$\theta_{2}$,are the$\eta$

and$\theta$

the duality, the V’-divergence

coordinates of points$D^{*}$$p_{1}$

and,$p_{2}$

respectively. According to.

For instance, let $M$ be a set of positive probability distributions on a set$\nabla$asis given as$D^{*}(p_{1}||p_{2})=D(p_{2}||p_{1})$metric,$D$$\mathcal{X},$$g$the Fisher

and V’ the exponential and mixture connections, respectively. Then the expo-

nential divergence $D$ is given by$D(p_{1}||p_{2})= \int_{\mathcal{X}}p_{1}(x)\log\frac{p_{1}(x)}{p_{2}(x)}dx$,

34
which is identical to the Kullback-Leibler divergence $K(p_{1},p_{2})$ . Note that our manner ofnaming of divergences is different from Amari’s one.
Next, we tackle the converse problem, i.e., let us construct a natural dualistic structure

for an arbitrary manifold $M$ on which a potential

is given, where$U(\theta)$$\theta=[\theta^{i}]$is a local

coordinate system of $M$ . In the following, we restrict ourselves to a domain $0$ in which thepotential
system$U(\theta)$$\eta=[\eta_{i}]$is a convex function with respect to . We first define another coordinate$\theta$and the corresponding potential$\eta_{j}=\partial_{j}U(\theta)$

Then$\theta^{j}=\partial^{j}V(\eta)$,by a Legendre transformation as$V(\eta)$$V( \eta)=\max\theta\in\Theta\{\theta^{i}\eta_{i}-U(\theta)\}$holds, and the pair$(\theta, \eta)$

satisfy the identity$U(\theta)+V(\eta)-\theta\cdot\eta=0$

The metric$\hat{g}$.

.

on $M$ is defined by$\hat{g}_{ij}=\partial_{i}\partial_{j}U(\theta)$.

This definition can be rewritten as$\hat{g}_{ij}=\frac{\partial\eta_{j}}{\partial\theta^{i}}$which readily leads to the relation$\hat{g}^{ij}=\frac{\partial\theta^{j}}{\partial\eta_{i}}=\partial^{i}\partial^{j}V(\eta)$

This indicates that the coordinate systems$\theta$and$\eta$.

are mutually dual with respect to$\hat{g}$inthe sense of Eq. (1). Further let us set$T_{ijk}=\partial_{i}\partial_{j}\partial_{k}U(\theta)$,

and define the -connection by$\alpha$$\Gamma_{ijk}^{\langle\alpha)}=[ij;k]-\frac{\alpha}{2}T_{ijk}$with $[ij;k]$ the Levi-Civita connection, then$\theta$and$\eta$,
become $\alpha=+1$ and $-1$ affine co-ordinates respectively, which can be affirmed by a straightforward computation. In this35
way, a dualistic structure
potential$U(\theta)$$(\hat{g}, \nabla^{\langle+1)}, \nabla^{(-1)})$. The $(+1)$ -divergence is defined as follows:$D^{\langle+1)}(p_{1},p_{2})$where$(\theta_{1}, \eta_{1})$on $M$ is derived in a natural manner from theand$(\theta_{2}, \eta_{2})$$=$$U(\theta_{2})+V(\eta_{1})-\theta_{2}\cdot\eta_{1}$$=$$U(\theta_{2})-U(\theta_{1})-(\theta_{2}-\theta_{1})\cdot\partial_{\theta}U(\theta_{1})$,are the dual affine coordinates of points $p_{1},p_{2}\in M$ , respectively.

Note that the point whose coordinates vanish corresponds to the minimum of the potential$\eta$$U(\theta)$

3.

Dualistic Dynamical Systems

In this section, we examine dualistic structures of a gradient system on a flat manifold.

Theorem 3.1 Let $M$ be a flat manifold with respect to the dualistic structure$U(p)$$(g, \nabla, \nabla^{*})$,

a potential function on $M$ with respect to an arbitmrily prefixed point $q\in M$ defined$by$$U(p)=D(q||p)$ ,where $D(q||p)$ is the$\nabla$-divergence. Then the gradient flow [7, p. 205]
(2)$\dot{\theta}^{i}=-g^{ij}\partial_{j}U(\theta)$

converges to the point
$p$

$q$

along the$\nabla^{*}$-geodesic, where$\theta$

is the$\nabla$-affine coordinates of point, and $U(\theta)=U(p(\theta))$ .

ProofSince V-divergence $D(q\Vert p)$ is rewritten as$D(q||p)$$=$$\psi(\theta(p))+\phi(\eta(q))-\theta(p)\cdot\eta(q)$$=$$\psi(\theta(p))+\{-\psi(\theta(q))+\theta(q)\cdot\eta(q)\}-\theta(p)\cdot\eta(q)$$=$$\psi(\theta(p))-\psi(\theta(q))+\{\theta(q)-\theta(p)\}\cdot\eta(q)$the gradient flow can be expressed in the form$\dot{\theta}^{i}(p)=-g^{ij}\{\partial_{j}\psi(\theta(p))-\eta_{j}(q)\}$

.

,

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By multiplying$g_{ji}$to both sides and using the identity$g_{ji} \dot{\theta}^{i}=\frac{\partial\eta_{j}}{\partial\theta^{i}}\frac{d\theta^{i}}{dt}=\frac{d\eta_{j}}{dt}$we have$\dot{\eta}_{j}(p)=-\{\eta_{j}(p)-\eta_{j}(q)\}$,which can readily be integrated to obtain$\eta_{j}(p(t))=\eta_{j}(q)+\{\eta_{j}(p(0))-\eta_{j}(q)\}e^{-t}$.

This proves the proposition.

$\blacksquare$

Example 3.1 Here we give two examples of Theorem 3.1. Let us consider a Gaussianfamily with mean$\mu$

and variance$\sigma^{2}$:$p_{\theta}(x)= \frac{1}{\sqrt{2\pi}\sigma}\exp[-\frac{(x-\mu)^{2}}{2\sigma^{2}}]$.

This is a typical example of exponential family since it can be represented in the form$\log p_{\theta}(x)=\theta^{1}f_{1}(x)+\theta^{2}f_{2}(x)-\psi(\theta)$

where$\theta^{2}=\frac{1}{2\sigma^{2}}$$\theta^{1}=\frac{\mu}{\sigma^{2}’}$are e-affine pammeters and$f_{1}(x)=x$ ,Throughout this example,

We first let

us set$\nabla$$g$$f_{2}(x)=-x^{2}$,$\psi(\theta)=\frac{\mu^{2}}{2\sigma^{2}}+\log\sqrt{2\pi}\sigma$.

is the Fisher metric.

and V’ be exponential and mixture connections, respectively. Further let

as a -distribution concentrated on the origin. Then the potential becomes$\delta$$q$$U(\theta)=D^{(e)}(q||p_{\theta})=K(q,p_{\theta})=\psi(\theta)$,

and the corresponding gradient flow coincides with Nakamura’s dynamics [8], which converges to the -distribution$\delta$$q$along an m-geodesic.

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Conversely, let$\nabla$and$\nabla^{*}$

be mixture and exponential connections, respectively. Further

let us set as a uniform distribution on$q$$\mathcal{X}$, then$\theta^{2}(q)$vanishes and-affine pammeters are the expectation pammete
denotes expectation at , and the dynamics takes the form

In this case,$\nabla$$\theta^{1}(q)$remains indefinite.

$rs\eta_{i}=E_{\theta}[f_{i}(x)]$where$E_{\theta}[\cdot]$$p_{\theta}$$\dot{\eta}_{i}=-g_{ij}\Psi U(\eta)$.

(3)

Since the potential becomes$U(\eta)=D^{(m)}(q||p_{\theta})=K(p_{\theta}, q)=-$[ $entropy$ of$p_{\theta}$] $+const.$ ,the dynamics is a steepest ascent flow of entropy, which converges to the uniform distribu-

tion

$q$

along an e-geodesic. Moreover, if we rescale the time logarithmically such as$t\dot{\eta}_{i}=-g_{ij}\partial^{j}U(\eta)$,(4)then the dynamics can be integmted easily and expressed in the e-affine pammeters as,$\theta^{j}(t)=\theta^{j}(q)+\frac{\theta^{j}(0)-\theta^{j}(q)}{t}$where$\theta^{j}(0)$also in theis the e-affine coordinates of the initial point. This solution can be expressed$(\mu, \sigma)$space as$\mu(t)$$=$$\frac{\theta^{1}(t)}{2\theta^{2}(t)}=\frac{\theta^{1}(0)-\theta^{1}(q)}{2\theta^{2}(0)}+\frac{\theta^{1}(q)}{2\theta^{2}(0)}t$$\sigma^{2}(t)$$=$$\frac{1}{2\theta^{2}(t)}=\frac{1}{2\theta^{2}(0)}t$,.

Here we used the relation $\theta^{2}(q)=0$ . If we set$\mu_{0}=\frac{\theta^{1}(0)-\theta^{1}(q)}{2\theta^{2}(0)}$,$v= \frac{\theta^{1}(q)}{2\theta^{2}(0)}$$D= \frac{1}{4\theta^{2}(0)}$then we have$\mu(t)=\mu_{0}+vt$,$\sigma^{2}(t)=2Dt$,

which shows that the dynamics (4) is nothing but a Uhlembeck-Ornstein process [9].
Next, we consider another situation. Given a manifold $M$ and alocally convex potential$U(\theta)$, then we can induce a natural dualistic structure$(\hat{g}, \nabla^{\langle+1)}, \nabla^{(-1)})$

on $M$ by the38procedure mentioned in the previous section. Let us examine a gradient flow on $M$ of theform$\dot{\theta}^{i}=-\hat{g}^{ij}\partial_{j}U(\theta)$,(5)

which can be reexpressed in the dual affine coordinates as$\dot{\eta}_{t}=-\eta_{i}$.

(6)In case $\dim M$ is even, this dynamical system can be characterized as a completely integrable

Hamiltonian system [1, p. 392], which is a generalization of Nakamura’s results [8], asfollows.
Theorem 3.2 If $\dim M$ is eve , say $2m$ , then the dynamical system (6) is a completely$n$integrable Hamiltonian system with position $Q_{k}=\eta_{2k}$ , momentum$P^{k}=-1/\eta_{2k-1}$, andHamiltonian $H=-Q_{k}P^{k},$ $(k=1, \cdots, m)$ . The $m$ quantities $?f_{k}=\eta_{2k}/\eta_{2k-1}$ are mutuallyindependent constants of motion.

Proof

By using (6), we have$\dot{\mathcal{H}}_{k}=\frac{1}{\eta_{2k-1}^{2}}(\dot{\eta}_{2k}\eta_{2k-1}-\eta_{2k^{\dot{7}}}\rho_{k-1})=0$

Independency and involutiveness of$\{H_{k}\}_{k=1}^{m}$.

are trivial. By straightforward computation,

Hamilton’s equations$\frac{dQ}{dt}=\frac{\partial H}{\partial P^{k}}$$\frac{dP}{dt}=-\frac{\partial \mathfrak{X}}{\partial Q_{k}}$are reduced to$\dot{r}_{hk}=-\eta_{2k}$,$\dot{\eta}_{2k-1}=-\eta_{2k-1}$

which reproduce the original gradient flow (6).

,$\blacksquare$

Note that if $\dim M$ is odd, then the dynamical system (6) can be regarded as a subdy-

namics of a higher dimensional completely integrable Hamiltonian system by combining itwith an independent odd dimensional gradient system.

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4Constrained Dynamics on a Parametric ModelIn this section, we examine a dynamical system which is induced on a parametric statisticalmodel. Let$M=\{p_{\theta}\}_{\theta\in\Theta}$

distributions$\varphi$on$\mathcal{X}$be a parametric model embedded in the set of probability. Theorem 3.1 indicates that the gradient flow in

potential $U(p)=K(q_{n},p)$ with

$q_{n}$

with respect to thethe empirical distribution is a dynamical system whose

gradient vector is m-tangent vector from the point$q_{n}$$\varphi$toward the empirical distribution$p$which in general falls out of the model $M$ . Therefore we can construct a constrained

dynamics on the model $M$ by projecting the gradient m-tangent vector onto the tangent

space $T_{p}(M)$ of the model with respect to the Fisher metric.
Theorem 4.1 Such an induced dynamical system is also a gradient flow on $M$ of the form$\dot{\theta}^{i}=-g^{ij}\partial_{j}K(q_{n},p_{\theta})$

where

$g$

,

(7)

is the Fisher metric on M. This flow converges to a locally maximum likelihood

estimate.
Proof

Let us define a bilinear form$\langle\cdot, \cdot\rangle$

on $T_{p}(M)$ by$\langle f(x),g(x)\rangle=\int_{\mathcal{X}}f(x)g(x)dx$,

where $f(x)$ and $g(x)$ are an m-tangent vector and an $e-tangent.vector$ , respectively. Notethat this value is identical to the conventional Fisher inner product in information geometry. Then the projection of the m-tangent vector from the point

distribution$q_{n}$

onto the tangent space $T_{p}(M)$ , expressed as$\langle q_{n}(x)-p_{\theta}(x),$$\partial_{j}\log p_{\theta}(x)$}$a^{i}\partial_{i}p_{\theta}(x)$toward the empirical, satisfies$=\langle a^{i}\partial_{i}p_{\theta}(x), \partial_{j}\log p_{\theta}(x)\rangle$

This leads to$a^{i}$$p$$=$$g^{ij}\langle q_{n}(x)-p_{\theta}(x), \partial_{j}\log p_{\theta}(x)\rangle$$=$$g^{ij}\langle q_{n}(x), \partial_{j}\log p_{\theta}(x)\rangle$$=$$-g^{ij}\partial_{j}K(q_{n},p_{\theta})$

.

.

40
Hence the induced dynamical system becomes$\dot{p}_{\theta}(x)=\dot{\theta}^{i}\partial_{i}p_{\theta}(x)=a^{i}\partial_{i}p_{\theta}(x)$,or$\dot{\theta}^{i}=a^{i}=-g^{ij}\partial_{j}K(q_{n},p_{\theta})$.

Every equilibrium point of this flow satisfies $a^{i}=0$ for all , which is nothing but foots of$i$the m-geodesic perpendiculars from
Lemma 4.1 Suppose the potential$i.e.,$$U(\theta)=K(q_{n},p_{\theta})$for every$q_{n}\in\varphi$

Proof

If

onto the model $M$ .

$U(\theta)$$M$

is identical to$\blacksquare$

on $M$ is given by the Kullback-Leibler divergence ,. The induced metric

iff the model$\hat{g}_{ij}(\theta)$$q_{n}$$\hat{g}_{ij}(\theta)$

is identical to the Fisher metricis an exponential family.

$g_{ij}(\theta)$

for every$q_{n}\in\varphi$

then$g_{ij}( \theta)-\hat{g}_{ij}(\theta)=-\int_{\mathcal{X}}\{p_{\theta}(x)-q_{n}(x)\}\partial_{i}\partial_{j}\log p_{\theta}(x)dx=0$This shows that$\partial_{i}\partial_{j}\log p_{\theta}(x)$$g_{ij}(\theta)$.

does not depend on , i.e., there exists a function$x$$\psi(\theta)$

such

that
$\partial_{i}\partial_{j}\log p_{\theta}(x)=-\partial_{i}\partial_{j}\psi(\theta)$

holds. This equation can readily be integrated to yield

Iog $p_{\theta}(x)=c(x)+\theta^{i}f_{i}(x)-\psi(\theta)$,which shows that the model$\{p_{\theta}(x)\}$is an exponential family. The converse statement isevident from the calculations above.

$\blacksquare$

Theorem 4.2 If model $M$ is an exponential family, then the induced gradient flow (7)converges to the unique maximum likelihood estimate with respect to the empirical distribution$q_{n}$

along m-geodesic. Moreover, if $\dim M$ is even, say $2m$ , then the flow is acompletely integmble Hamiltonian system with position$P^{k}=-1/\partial_{2k-1}K(q_{n},p_{\theta})$Proof, and Hamiltonian$\mathcal{H}=-Q_{k}P^{k},$$Q_{k}=\partial_{2k}K(q_{n},p_{\theta})$$k=1,$$\cdots,$$m$

Straightforward from Theorems 3.1, 3.2, 4.1, and Lemma 4.1.

, momentum.
$\blacksquare$

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Concluding Remarks

5

We have constructed a gradient flows on a flat manifold $M$ with respect to a dualistic
structure$\dim M$$(g, \nabla, \nabla^{*})$

which converges to an arbitrarily prefixed point along V-geodesic. If

is even, this flow can be also characterized as a completely integrable Hamiltonianflow.

We have also derived a constrained dynamics on a submanifold $M$ embedded in astatistical manifold$\varphi$

which converges to the locally maximum likelihood estimate. If $M$

is an exponential family, then the flow evolves along m-geodesics. In case $\dim M$ is even,the flow can also be considered as a completely integrable Hamiltonian system. However,ststistical meaning of such characterization as a Hamiltonian system is not clear.
In a basic sense, a $2n$ dimensional Hamiltonian system is equivalent to a$n$dimensional

Lagrangean system. From this analogy, we can imagine a 2nd order dynamics of the form$\ddot{\theta}^{k}+t^{k}ij\}\dot{\theta}^{i}\dot{\theta}^{j}=-g^{kj}\partial_{j}U(\theta)$,

which is the equation of motion of a particle constrained on a manifold $M$ associated witha potential$U(\theta)$. It is well known that this dynamics can be derived by the variational

principle with Lagrangean$L= \frac{1}{2}g_{ij}\dot{\theta}^{i}\dot{\theta}^{j}-U(\theta)$.

In the same way, if we consider a dynamical system of the form$\ddot{\theta}^{k}+\Gamma_{ij}^{\langle-1)k}\dot{\theta}^{i}\dot{\theta}^{j}=-\hat{g}^{kj}\partial_{j}U(\theta)$,

then we have$\ddot{\eta}_{i}=-\eta_{i}$in the dual affine coordinates, which indicates that the system is composed of independent$n$harmonic oscillators and can be regarded as a completely integrable Hamiltonian system.
In this case, however, it is not clear whether the system can be derived by a certain
variational principle.

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Acknowledgments
Gratitude is expressed to Prof. S. Amari for his critical reading of the manuscript andvaluable comments. Thanks are due to Prof. Y. Nakamura for fruitful discussions.

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