Keywords

1 Introduction

This paper develops non-parametric statistical manifolds modelled on spaces of Sobolev type. It applies some of the results of [12] to a particular class of manifolds, and develops smoothly embedded finite-dimensional exponential submanifolds. The non-parametric manifolds are natural refinements of those in [10] and [11]; they employ charts that are “balanced” between the density function and its log. The inverses of the charts are expressed in terms of a deformed exponential function having linear growth, a property shared by other deformed exponentials (notably the Kaniadakis exponential with parameter \(\kappa =1\)). (See [8] and the discussion in [12]). The linear growth property is highly advantageous in the Sobolev context because the deformed exponential then “lifts” to a nonlinear superposition (Nemytskii) operator that acts continuously on particular classes of model spaces.

For some , let \(\mathcal{X}\) be the \(\sigma \)-algebra of Lebesgue measurable subsets of , and let \(\mu \) be a probability measure on \(\mathcal{X}\) that is mutually absolutely continuous with respect to Lebesgue (volume) measure. \(\mathcal{X}\) is a very rich collection of subsets, , for which the Lebesgue measure dx(A) is well defined. Each \(A\in \mathcal{X}\) has a well-defined probability \(\mu (A)\), which can be expressed in terms of the probability density function as follows:

$$\begin{aligned} \mu (A) = \int _A r(x)\,dx. \end{aligned}$$
(1)

The simplest example of a statistical manifold over the sample space is the finite-dimensional exponential model [1, 3]. This is based on a finite set of linearly independent random variables \(\eta _1,\ldots \eta _n\) defined on . Let B be an open subset of such that, for any \(y\in B\), \({{\mathbf{E}_\mu }\exp (\sum _i y_i\eta _i)<\infty }\), where \({\mathbf{E}_\mu }\) is expectation (integration) with respect to \(\mu \). Any \(y\in B\) represents the probability measure \(P_y\), defined by

$$\begin{aligned} P_y(A) = \int _A \exp \bigg (\sum _i y_i\eta _i-c\bigg )\mu (dx), \end{aligned}$$
(2)

where \({c=\log {\mathbf{E}_\mu }\exp (\sum _i y_i\eta _i)}\). The set \(N:=\{P_y: y\in B\}\) is a manifold of probability measures, with a differentiable structure in terms of which the important statistical divergences of estimation theory are suitably smooth.

The first fully successful infinite-dimensional (non-parametric) statistical manifold was constructed in [14], and further developed in [2, 5, 13]. This is the natural extension of exponential models such as N to the non-parametric setting. The chart is a centred version of the log of the probability density function \(p:=dP/d\mu \), and so, as in (2), p is represented in terms of the exponential of the model space variable. The model space used is the exponential Orlicz space, which has a stronger topology than the Lebesgue \(L^\lambda (\mu )\) spaces for \(1\le \lambda <\infty \).

A central requirement of a chart in a statistical manifold is that it should induce a topology with respect to which statistical divergences, such as the Kullback-Leibler (KL)-divergence, are appropriately smooth. The KL-divergence between finite measures P and Q on \(\mathcal{X}\) is defined as follows [1, 3]:

(3)

It is of class \(C^\infty \) on the exponential Orlicz manifold. As (3) shows, the KL-divergence is bilinear in the density p and its log, and so its smoothness properties are closely connected with those of p and \(\log p\) considered as elements of dual function spaces. This is why the following deformed logarithm was introduced in the Hilbert setting of [10]:

$$\begin{aligned} \log _d(y) = y - 1 + \log y. \end{aligned}$$
(4)

This is composed with probability density functions to realise a chart on a manifold of finite measures that maps into the Lebesgue space \(L^\lambda (\mu )\), for any \(2\le \lambda <\infty \) [10, 11]. A centred version of this can be used as a chart on the submanifold of probability measures. The inverse of \(\log _d\) can be thought of as a deformed exponential function. It has linear growth, as a result of which the density, p, and its log both belong to the same space as \(\log _d(p)\) (i.e. the model space \(L^\lambda (\mu )\)). This property is not shared by the exponential Orlicz manifold. Reference [12] shows that it is retained when the sample space is and the model space \(L^\lambda (\mu )\) is replaced by particular spaces of Sobolev type.

The natural domain of statistical divergences, such as the KL-divergence, is a space of measures defined on an abstract measurable space \((\varOmega ,\mathcal{F})\). Since the primary concern of “raw” information geometry is the smoothness of these divergences, the exponential Orlicz and \(L^\lambda (\mu )\) manifolds of [14] and [11], in their general form, make no reference to any other structures that the sample space \(\varOmega \) may possess. However, in the special case that , the topology and linear structure of play important roles in many applications. For example, the Fokker-Planck equation makes direct reference to the linear structure of through a differential operator. For this reason, it is of interest to develop “hybrid” information manifolds, in which the topology of the sample space is somehow incorporated into the model space. One way of achieving this is to use model spaces of Sobolev type. This approach is taken here, in the context of the \(L^\lambda (\mu )\) manifolds of [10, 11]. For the development of Sobolev variants of the exponential Orlicz manifold, the reader is referred to [7].

The paper is structured as follows. Section 2 introduces the spaces on which the manifolds are modelled. Section 3 presents the principal results on the non-parametric manifolds constructed from these spaces; it discusses both manifolds of finite measures and submanifolds of probability measures. Finally Sect. 4 develops a class of smoothly embedded finite-dimensional exponential manifolds that are of potential use in applications.

2 The Model Spaces

For some \(t\in (1,2]\), let \(\theta _t:[0,\infty )\rightarrow [0,\infty )\) be a strictly increasing, convex function that is twice continuously differentiable on \((0,\infty )\), such that \(-\sqrt{\theta _t}\) is convex, \(\lim _{z\downarrow 0}\theta _t^\prime (z)<\infty \), and

(5)

Examples, including some for which is of class \(C^2\) are given in [12], which also develops variants for which \(t\in (0,1]\). (The restriction \(t\in (1,2]\), used here, simplifies the presentation in what follows while retaining a useful subset of the manifolds developed in [12]). For each \(t\in (1,2]\), we define a reference probability measure, \(\mu =\mu _t\), as follows.

$$\begin{aligned} \mu _t(dx) = \exp (l_t(x))dx, \quad \mathrm{where\ \ } l_t(x):=\textstyle \sum _i (C_t-\theta _t(|x_i|)), \end{aligned}$$
(6)

and is such that \(\int \exp (C_t-\theta _t(|z|))dz=1\). For any \(1\le \lambda <\infty \), let \(L^\lambda (\mu )\) be the Banach space of (equivalence classes of) measurable functions for which \(\Vert u\Vert _{L^\lambda (\mu )}:=(\int |u|^\lambda d\mu )^{1/\lambda }<\infty \).

For , let \(S:=\{0,\ldots , k\}^d\) be the set of d-tuples of integers in the range \(0\le s_i\le k\). For \(s\in S\), we define \(|s|=\sum _is_i\), and denote by 0 the d-tuple for which \(|s|=0\). For any \(0\le i\le k\), \(S_i:=\{s\in S:i\le |s|\le k\}\) is the set of d-tuples of weight at least i and at most k. Let \(\varLambda =(\lambda _0,\lambda _1,\ldots ,\lambda _k)\), where \(\lambda _i\in [1,\infty )\) for \(0\le i\le k\), and let \(W^{k,\varLambda }(\mu )\) be the mixed-norm, weighted Sobolev space comprising functions \(a\in L^{\lambda _0}(\mu )\) that have weak partial derivatives \(D^sa\in L^{\lambda _{|s|}}(\mu )\), for all \(s\in S_1\). For \(a\in W^{k,\varLambda }(\mu )\) we define the mixed norm

$$\begin{aligned} \Vert a\Vert _{W^{k,\varLambda }(\mu )} := \bigg (\sum _{s\in S_0}\Vert D^sa\Vert _{L^{\lambda _{|s|}}(\mu )}^{\lambda _0} \bigg )^{1/\lambda _0} < \infty . \end{aligned}$$
(7)

\(W^{k,\varLambda }(\mu )\) is a Banach space with respect to this norm. (See Theorem 2.1 in [12]).

We shall confine our attention here to the following special class of model spaces, parametrised by and \(\lambda \in [k,\infty )\):

(8)

This includes the Hilbert-Sobolev space \(G_m^{1,2}\). Let be the inverse of the deformed logarithm of (4). The following is proved as part of Proposition 2 in [12].

Proposition 1

(i):

For any \(a\in G_m^{k,\lambda }\), \(\psi (a)\in G_m^{k,\lambda }\).

(ii):

 The nonlinear superposition (Nemytskii) operator \(\varPsi _m^{k,\lambda }:G_m^{k,\lambda }\rightarrow G_m^{k,\lambda }\), defined by \(\varPsi _m^{k,\lambda }(a)(x)=\psi (a(x))\), is continuous.

This is rare property in the theory of nonlinear maps between Sobolev spaces, and has its origins in the boundedness of the derivatives of \(\psi \). It is useful in the construction of finite-dimensional mixture or exponential submanifolds. Normally the Sobolev space forming the domain of a continuous nonlinear superposition operator would need a stronger topology than that forming its range [15]. This would require it to have larger Lebesgue exponents, or to control a greater number of derivatives.

Remark 1

It is shown in [12] that the continuity of \(\varPsi \) between identical Sobolev spaces is also true of the fixed-norm space \(W^{2,(1,1,1)}(\mu )\), but of no other fixed-norm spaces (except for \(G_m^{1,\lambda }\) with \(\lambda \in [1,\infty )\)).

3 The Nonparametric Manifolds

Let \(G=G_m^{k,\lambda }\) be the mixed-norm Sobolev space as defined in (8). Let \(\varPsi :=\varPsi _m^{k,\lambda }\) be as defined in Proposition 1. We consider the set M of finite measures on \(\mathcal{X}\) satisfying the following:

(M1):

P is mutually absolutely continuous with respect to \(\mu \);

(M2):

\(p, \log p\in G\), where \(p=dP/d\mu \).

This is equipped with the global chart \(\phi :M\rightarrow G\), defined by

$$\begin{aligned} \phi (P) = \log _d p = p - 1 +\log p. \end{aligned}$$
(9)

In view of Proposition 1, it is not difficult to show that \(\phi \) is a bijection onto G. (See Proposition 1 in [12]). For any \(P\in M\), let \({\tilde{P}}_a\in M\) have density \(d{\tilde{P}}_a/d\mu =\psi ^{(1)}(a)\), where \(a=\phi (P)\) and \(\psi ^{(1)}=\psi /(1+\psi )\) is the first derivative of \(\psi \). We define a tangent vector, U at \(P\in M\), to be a signed measure on \(\mathcal{X}\) of finite total variation such that

(T1):

U is mutually absolutely continuous with respect to \({\tilde{P}}_a\);

(T2):

\(dU/d{\tilde{P}}_a\in G\), where \(a=\phi (P)\).

The tangent space \(T_PM\) is the linear space of all such signed measures, and the tangent bundle is the disjoint union \(TM=\cup _{P\in M}(P,T_PM)\). This admits the global chart \(\varPhi :TM\rightarrow G\times G\), defined by

$$\begin{aligned} \varPhi (P,U) = (\phi (P), dU/d{\tilde{P}}_{\phi (P)}). \end{aligned}$$
(10)

The derivative of a (Fréchet) differentiable, Banach-space-valued map (at P and in the “direction” U) is defined as follows: (clearly \(u=U\phi \)).

$$\begin{aligned} Uf = (f\circ \phi ^{-1})_a^{(1)}u, \quad \mathrm{where\ }(a,u)=\varPhi (P,U). \end{aligned}$$
(11)

Let \(m_\lambda ,e_\lambda :M\rightarrow L^\lambda (\mu )\) be the nonlinear superposition operators defined by

$$\begin{aligned} m_\lambda (P)(x) = p(x)-1 \quad \mathrm{and}\quad e_\lambda (P)(x) = \log p(x). \end{aligned}$$
(12)

The map \(m_\lambda \) is the composition of \(\varPsi _m^{k,\lambda }-1\) with the inclusion map \(\imath :G\rightarrow L^\lambda (\mu )\). It is smoother than \(\varPsi _m^{k,\lambda }\) since its range has a weaker topology. The following is a corollary of Lemma 4 in [12].

Lemma 1

\(m_\lambda ,e_\lambda \in C^1(M;L^\lambda (\mu ))\).

The smoothness properties of the KL-divergence on manifolds modelled on \(L^\lambda (\mu )\) is investigated in detail in [11]. Its derivatives can be used to construct the Fisher-Rao metric and Amari-Chentsov tensor on M by the Eguchi method [4]. The Fisher-Rao metric is the covariant 2-tensor field defined, for \(\lambda \ge 2\), by

$$\begin{aligned} {\langle U, V\rangle _P = {\mathbf{E}_\mu }Um_\lambda Ve_\lambda = {\mathbf{E}_\mu }Ue_\lambda Vm_\lambda = {\mathbf{E}_\mu }\frac{p}{(1+p)^2}U\phi V\phi .} \end{aligned}$$
(13)

The Amari-Chentsov tensor is the covariant 3-tensor field defined, for \(\lambda \ge 3\), by

$$\begin{aligned} {\tau _P(U,V,W) = {\mathbf{E}_\mu }Um_\lambda Ve_\lambda We_\lambda = {\mathbf{E}_\mu }\frac{p}{(1+p)^3}U\phi V\phi W\phi } \end{aligned}$$
(14)

Corollary 1

(i):

If \(\lambda \ge 2\) then the Fisher-Rao metric is a continuous covariant 2-tensor field on M.

(ii):

If \(\lambda \ge 3\) then the Amari-Chentsov tensor is a continuous covariant 3-tensor field on M.

Proof

Both parts follows from the first representations in (13) and (14), Lemma 1 and the chain rule of Fréchet derivatives.    \(\square \)

In the raw (non-Sobolev) Hilbert manifold of [10], the composition map is continuous for all continuous vector fields \(\mathbf{U}\), \(\mathbf{V}\). However, the metric is not continuous in the stronger “operator topology” of Corollary 1(i). The extra regularity here arises from the log-Sobolev embedding theorem, and is not retained if \(t\in (0,1]\). (See Lemma 4 in [12]). Similarly, in the raw Banach manifold of [11] with \(\lambda \ge 3\), the composition map is continuous for all continuous vector fields \(\mathbf{U}\), \(\mathbf{V}\), \(\mathbf{W}\), but not continuous in the sense of Corollary 1(ii) unless \(\lambda >3\).

Let \(M_0\subset M\) be the subset of M whose members are probability measures. These satisfy the additional hypothesis:

(M3):

\({\mathbf{E}_\mu }p = 1\).

The co-dimension 1 subspace of G whose members, a, satisfy \({\mathbf{E}_\mu }a=0\) will be denoted \(G_0\). Let \(\phi _0:M_0\rightarrow G_0\) be defined by

$$\begin{aligned} \phi _0(P) = \phi (P) - {\mathbf{E}_\mu }\phi (P) = \log _d p - {\mathbf{E}_\mu }\log _d p. \end{aligned}$$
(15)

The following is a special case of parts of Propositions 5 and 6 in [12].

Proposition 2

(i):

\((M_0,G_0,\phi _0)\) is a \(C^{\lfloor \lambda \rfloor }\)-embedded submanifold of \((M,G,\phi )\).

(ii):

In terms of the charts \(\phi _0\) and \(\phi \), the inclusion map \(\rho :G_0\rightarrow G\) has the following form

$$\begin{aligned} \rho (a) = a + Z(a), \end{aligned}$$
(16)

where is an (implicitly defined) normalisation function.

(iii):

The first and (if \(\lambda \ge 2\)) second derivatives of \(\rho \) are as follows:

(17)

where , \({\tilde{P}}_a\) is the finite measure defined after (9) and \(\psi ^{(i)}\) is the ith derivative of \(\psi \).

For any \(P\in M_0\), tangent vectors \(U\in T_PM_0\) are distinguished from those only in \(T_PM\) by the fact that . The pushforward of the inclusion map \(\imath :M_0\rightarrow M\) splits \(T_PM\) into \(T_PM_0\) and the complementary subspace of signed measures .

Remark 2

The probability measure \(P_a\) in (17) is the escort probability in the interpretation of \(M_0\) as a deformed exponential model [9].

4 Finite-Dimensional Exponential Models

The \(\alpha \)-divergences (and their derivatives such as the Fisher-Rao metric) are at least as smooth on the Sobolev manifolds of Sect. 3 as they are on their non-Sobolev counterparts (as developed in [10, 11]) because the Sobolev manifolds have stronger topologies. When it comes to embedded submanifolds, however, this benefit is reversed. Theorem 5.1 in [10] shows that any finite dimensional exponential manifold that is contained in the raw Hilbert manifold of [10] is smoothly embedded in that manifold. That this is not so of the Hilbert-Sobolev manifold M modelled on \(G_m^{1,2}\) is demonstrated by the following example.

Example 1

Let \(d=1\), let \(\mu \) be the standard Gaussian measure, \(k=1\) and \(\lambda =2\). For \(i=0,1\), let \(p_i:=\exp (\eta _i)\) where \(\eta _0=3x^2/16\) and \(\eta _1=\sin (\exp (3x^2/16))\). Then \(P_0\) and \(P_1\) are both in M, but the measure with density \(\exp ((\eta _0+\eta _1)/2)\) is not, since its derivative is not square integrable.

Nevertheless, the smooth embedding property can be recovered under additional hypotheses. Let \(G=G_m^{k,\lambda }\) be the general mixed-norm space of Sect. 2, and let M be the associated manifold of finite measures. Since the Fisher-Rao metric is positive definite on M, it is a (strong) Riemannian metric on any finite-dimensional, smoothly embedded submanifold, N, and the full geometry of dual \(\pm \alpha \)-covariant derivatives (for \(\alpha \in [-1,1]\)) is realised on N.

For some , let \(1,\eta _1,\ldots ,\eta _n\) be linearly independent members of G, and for any let P(y) be the measure on \(\mathcal{X}\) with density

$$\begin{aligned} p(y) := \exp \gamma (y), \quad \mathrm{where}\quad \gamma (y) = \sum _{j=0}^n y_j\eta _j\quad \mathrm{and}\quad \eta _0\equiv 1. \end{aligned}$$
(18)

The function is clearly injective. Let be open and such that \(P(y)\in M\) for every , and let . As well as being a subset of M, N is a finite-dimensional exponential model with chart , defined by \(\theta =\gamma ^{-1}\circ e_\lambda \circ \imath \), where \(e_\lambda \) is as defined in (12) and \(\imath :N\rightarrow M\) is the inclusion map.

Theorem 1

Suppose that \(\lambda \ge \max \{2,k\}\) and that, for every \(y\in B\), \(1\le j\le n\) and \(s\in S_1\),

$$\begin{aligned} {\mathbf{E}_\mu }\left| p(y)D^s\eta _j\right| ^{\lambda /|s|} < \infty ; \end{aligned}$$
(19)

then N is a \(C^\infty \)-embedded submanifold of M.

Proof

A partition of \(s\in S_1\) is a set \(\pi =\{\sigma _1,\ldots ,\sigma _n\in S_1\}\) such that \(\sum _i\sigma _i=s\). Let \(\varPi (s)\) denote the set of partitions of s. According to the Faá di Bruno formula, for any \(s\in S_1\),

$$\begin{aligned} D_x^sp(y) = p(y)\sum _{\pi \in \varPi (s)}K_\pi \prod _{\sigma \in \pi }D_x^\sigma \gamma (y), \end{aligned}$$
(20)

where the \(K_\pi <\infty \) are combinatoric constants, and x is made explicit in \(D_x^s\) for the sake of clarity.

As in the proof of Theorem 5.1 in [10], we define a local coordinate system around a generic . Let \(\epsilon >0\) be such that the ball of centre y and radius \(\epsilon \) is contained in , and let B(y, r) be the ball of centre y and radius r. For any \({\tilde{y}}\in B(y,\epsilon /2n)\) let \(\zeta \in (1/4,3/4)^{n+1}\) be defined by \(\zeta _j=(1+(n+1)\epsilon ^{-1}({\tilde{y}}-y)_j)/2\); then

$$ {\tilde{y}}= \frac{1}{n+1}\sum _{j=0}^n\big ((1-\zeta _j)(y-\epsilon \mathbf{e}_j) +\zeta _j(y+\epsilon \mathbf{e}_j)\big ), $$

where is the coordinate orthonormal basis. Differentiating p with respect to y, for any ,

$$\begin{aligned} D_y^\alpha p({\tilde{y}}) = (2\epsilon )^{-|\alpha |}\prod _{j=0}^n\left( p_{j-}^{1-\zeta _j}p_{j+}^{\zeta _j} \log ^{(n+1)\alpha _j}(p_{j+}/p_{j-})\right) ^{1/(n+1)}, \end{aligned}$$
(21)

where \(p_{j\pm }=p(y\pm \epsilon \mathbf{e}_j)\). The product rule now shows that, for any \({\tilde{y}}\in B(y,\epsilon /2n)\),

$$\begin{aligned} D_y^\alpha D_x^sp({\tilde{y}}) = \sum _{\beta \le \alpha }K_\beta D_y^{\alpha -\beta }p({\tilde{y}}) \sum _{\pi \in \varPi (s)}K_\pi D_y^\beta \prod _{\sigma \in \pi }D_x^\sigma \gamma ({\tilde{y}}), \end{aligned}$$
(22)

where the \(K_\beta <\infty \) are combinatoric constants. For any there is a \(K_m<\infty \) such that, for all \(q,r\in (0,\infty )\) and all \(\delta \in (1/4,3/4)\),

$$ q^{1-\delta }r^\delta |\log (q/r)|^m = \frac{q+r}{(q/r)^\delta +(r/q)^{1-\delta }}|\log (q/r)|^m \le K_m(q+r). $$

Applying this to (21) and (22), we obtain the bound

$$ \left| D_y^\alpha D_x^sp({\tilde{y}})\right| \le K\sum _{\beta \le \alpha }\sum _{\pi \in \varPi (s)}\prod _{j=0}^n\bigg |(p_{j-}+p_{j+}) D_y^\beta \prod _{\sigma \in \pi }D_x^\sigma \gamma ({\tilde{y}})\bigg |^{1/(n+1)}, $$

for some \(K<\infty \). It follows from (19) and Hölder’s inequality that the term, whose absolute value is taken on the right-hand side here, belongs to \(L^{\lambda /|s|}(\mu )\), and a further application of Hölder’s inequality shows that

$$\begin{aligned} {\mathbf{E}_\mu }\sup _{{\tilde{y}}\in B(y,\epsilon /2n)}\left| D_y^\alpha D_x^sp({\tilde{y}}) \right| ^{\lambda /|s|} < \infty . \end{aligned}$$
(23)

A Taylor expansion of \(D_y^\alpha D_x^s p_y\) about y, in the direction \(\mathbf{e}_j\), yields

$$ D_y^\alpha D_x^s p(y+t\mathbf{e}_j) = D_y^\alpha D_x^s p(y) + D_y^{\alpha +\mathbf{e}_j}D_x^s p(y)t + D_y^{\alpha +2\mathbf{e}_j}D_x^s p(y+\delta t\mathbf{e}_j)t^2/2, $$

for some \(\delta =\delta (y,t,j,x)\in [0,1]\). Together with (23) and the dominated convergence theorem, this shows that \((-\epsilon /2n,\epsilon /2n)\ni t\mapsto D_y^\alpha D_x^s p(y+t\mathbf{e}_j)\in L^{\lambda _{|s|}}(\mu )\) is differentiable at \(t=0\), with derivative \(D_y^{\alpha +\mathbf{e}_j}D_x^s p(y)\). An inductive argument thus establishes the infinite differentiability of . The same is clearly true of , and so we have shown that the inclusion map \(\imath \) is of class \(C^\infty \).

Expressed in terms of the charts, \(\imath \) takes the form \(f=\phi \circ e_\lambda ^{-1}\circ \gamma \). Let be defined by \(g=\pi \circ e_\lambda \circ \phi ^{-1}\), where is the \(L^2(\mu )\)-projection onto the subspace spanned by \((1,\eta _1,\ldots ,\eta _n)\). It follows from Lemma 1 that g is of class \(C^1\). Now \(g\circ f\) is the identity function of , and so f is a homeomorphism onto its image (endowed with the subspace topology), and its derivative, \(f^{(1)}\), is a toplinear isomorphism onto its image. So is a finite-dimensional closed linear subspace of both G and \(L^2(\mu )\). Let \(H^c\) be its orthogonal complement in \(L^2(\mu )\); then \(f^{(1)}\) splits G into the components and \(G\cap H^c\). So f is an immersion and an embedding. (See Proposition 2.3 in [6]). This completes the proof.    \(\square \)

Condition (19) is clearly satisfied if \(p(y)\in L^\infty (\mu )\) for all ; this is so, for example, if N is the exponential manifold of all non-singular (scaled) Gaussian measures on \(\mathcal{X}\), and \(t<2\). However, there are other possibilities, for example that in which \(\gamma (y)\), p(y) and their x derivatives have sub-exponential growth in x for all \(y\in B\).

Let \(N_0:=M_0\cap N\) be the subset of probability measures. This is, itself, a finite dimensional exponential manifold with chart \(\theta _0:N\rightarrow B\) defined by \(\theta _0(P)=(\theta (P)_1,\ldots ,\theta (P)_n)\).

Corollary 2

Under the hypotheses of Theorem 1:

(i):

\(N_0\) is a \(C^\infty \)-embedded submanifold of N;

(ii):

\(N_0\) is a \(C^\infty \)-embedded submanifold of \(M_0\).

Proof

The map was shown, in the proof of Theorem 1, to be of class \(C^\infty \). Its first derivative with respect to \(y_0\) is \({\mathbf{E}_\mu }p(y)\). Since this is strictly positive on , the implicit function theorem shows that , defined by \(\exp (f(z)+\sum _{j=1}^nz_j\eta _j)=1\), is of class \(C^\infty \). In terms of the charts \(\theta \) and \(\theta _0\), the inclusion map \(\imath :N_0\rightarrow N\) takes the form \(\varphi (z)=(f(z),z)\). This is clearly a \(C^\infty \)-embedding, which proves part (i).

Let \(\tau :G\rightarrow G_0\) be defined by \(\tau (a)=a-{\mathbf{E}_\mu }a\). This is of class \(C^\infty \), and so the same is true of the map \(g:=\tau \circ \phi \circ \imath \circ \theta _0^{-1}:B\rightarrow G_0\), where \(\imath :N_0\rightarrow M\) is the inclusion map. Now g is injective and, since \(\tau ^{(1)}\rho ^{(1)}\) is the identity map of \(G_0\), the same is true of its first derivative, at all points in B. The latter clearly splits \(G_0\), and so the inclusion map \(\imath :N_0\rightarrow M_0\) (which is expressed in charts by g) is an embedding. This completes the proof of part (ii).    \(\square \)