Immersions into Statistical Manifolds
aa r X i v : . [ m a t h . DG ] M a r Affine Immersions and Statistical Manifolds
Mahesh T V, K S Subrahamanian Moosath ∗ Department of MathematicsIndian Institute of Space Science And TechnologyValiyamala(po), Thiruvanathapuram, kerala, India- 695547
Abstract
In this paper first we discuss about statistical manifolds and immersed hyper surfaces in R n +1 . We show that everycentro-affine hyper surface is a statistical manifold realized in R n +1 . Then proved that dualy flat statistical manifoldcan be immersed as a centro- affine hyper surface. Also observed that a Blaschke immersed manifold is a statisticalmanifold. Then in the case of the dualy flat statistical manifold ( S , ∇ e , ∇ m ) we show that if all ∇ e - autoparallelsubmanifolds are exponential then S is an exponential family. Also prove that submanifold of a statistical manifoldwith ∇ e connection is ∇ e - autoparallel if it is exponential family. Keywords: statistical manifold, affine immersion,exponential family
Introduction
Information geometry investigates the differential geometric structure of statistical manifolds. It has got a widevariety of applications in the areas of engineering and science. Parametrized family of probability measures on asample space (finite or infinite) has got an affine space structure introduced by Amari and Chensov. They have alsogiven a family of affine connections with dualistic property on the parametrized family of probability measures. Soa family of parametrized probability density functions is a smooth manifold, called the statistical manifold, with theparameters as coordinates, Fisher information is the Riemannian metric and we have a family of affine connectionswith dualistic property. In the statistical manifold in which parameters are coordinate, the affine structure enables thenice representation of probability measures through the vector space of random variables. A statistical manifold isalso viewed as a Riemannian manifold with a three tensor which is symmetric (Lauritzen). Kurose [3, 4] studied theaffine immersions of statistical manifolds and Dillen, Nomizu , Vranken [5] studied affine hyper surfaces which hasrelevance in simply connected statistical manifold theory.In this paper we look at the affine geometric aspects of statistical manifolds and also of the ∇ e autoparallel sub-manifolds of exponential family. In section we recall the basic results in affine differential geometry. In section (2) ∗ Corresponding author
Email addresses: [email protected] (Mahesh T V), [email protected] (K S Subrahamanian Moosath )
Preprint submitted to Journal of Statistics and Probability Letter March 8, 2018 e introduce the notion of statistical manifold realized in R n +1 , there we show that every centro- affine hypersurfaceis a statistical manifold realized in R n +1 . Dually flat statistical manifold is locally- immersed as a centro-affine hy-persurface, also we observed that for a Blaschke immersion f : M −→ R n +1 , ( M , ∇ , h ) is a statistical manifold.In section we discuss about exponential family. Family of probability measures is a subset of infinite dimensionalaffine subspace of positive measures up to scale. Parametrized models are surfaces in this affine space, among that thegeometry of exponential family is the simplest one. Exponential family with ± - connection play an important role ininformation geometry, Amari [6] proved that submanifold M of an exponential family S is exponential if and only if M is a ∇ e - auto parallel submanifold. We show that if all ∇ e - auto parallel submanifold of a parametrized model S is exponential then S is an exponential family. Also Shown that submanifold of a parametrized model S which is anexponential family is a ∇ e - auto parallel submanifold. Also observed that exponential family is locally immersed as acentro- affine hypersurface.
1. Affine Immersion
In this section we give a brief description about immersion of a manifold in R n +1 as a hyper surface.Let M be an n -dimensional smooth manifold, J ( M ) denote the set of all real valued smooth functions on M , τ ( M ) be the set of all smooth vector fields on M and ∇ be an affine connection on M . A vector field X ∈ τ ( M ) is called parallel on M if ∇ X Y = 0 , ∀ Y ∈ τ ( M ) and M is called flat with respect to ∇ if ∇ ∂ i ∂ j = 0 for all i, j , where ∂ i are the basic vector fields with respect to the local coordinates ( x , x , x ...x n ) . The torsion tensor field T ( X, Y ) and curvature tensor field R ( X, Y ) Z with respect to ∇ are defined as T ( X, Y ) = ∇ X Y − ∇ Y X − [ X, Y ] R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z The Ricci Tensor of type (0 , is defined by Ric ( Y, Z ) = trace { X −→ R ( X, Y ) Z } .A symmetric tensor field g of type (0,2) is called a non-degenerate metric on M , if g ( X, Y ) = , ∀ Y ∈ τ ( M ) then X = 0 . It is also called pseudo-Riemannian metric. If g is positive definite then we say g is a Riemannian metric. Anaffine connection ∇ is called a metric connection if ∇ g = , where ∇ g denote the covariant derivative of the tensorfield g . If ∇ is metric then we have Xg ( Y, Z ) = g ( ∇ X Y, Z ) + g ( Y, ∇ X Z ) for every X, Y, Z ∈ τ ( M ) . Definition.
Let ( M , ∇ ) be n -manifold with an affine connection ∇ . An immersion f : M −→ R n +1 is called anaffine immersion if there is a vector field ξ on M such that the following holds(a) T f ( x ) ( R n +1 ) = f ∗ ( T x ( M )) + span { ξ x } (b) D X f ∗ ( Y ) = f ∗ ( ∇ X Y ) + h ( X, Y ) ξ The vector field ξ which satisfies (a) is called the transversal vector field. h is a symmetric bilinear function on thetangent space T x ( M ) , called the affine fundamental form and D denote the standard connection on R n +1 .2ote that for an immersion f : M −→ R n +1 having a transversal vector field ξ on M there is a torsion freeinduced connection ∇ satisfying D X f ∗ ( Y ) = f ∗ ( ∇ X Y ) + h ( X, Y ) ξ . Also for all X ∈ τ ( M ) , ξ satisfies D X ξ = − f ∗ ( SX ) + α ( X ) ξ , where S is a tensor of type (1 , called affine the shape operator and α is a -from calledthe transversal connection from. Definition.
Let ω be a fixed volume element on R n +1 . For a hyper surface immersion f : M −→ R n +1 with transver-sal vector field ξ we define the induced volume element η on M as η ( X , X , X ...X n ) = ω ( X , X , X ...X n , ξ ) .Then we have ∇ X η = α ( X ) η for every X ∈ τ ( M ) . Theorem 1.1.
Let f : M −→ R n +1 be an affine immersion with transversal vector field ξ , ∇ be the the inducedconnection, h be the affine fundamental form, S be the affine shape operator and α be the the transversal connectionthen the following equations hold.1 R ( X, Y ) Z = h ( Y, Z ) SX − h ( X, Z ) SY (Gauss)2 ( ∇ X h )( Y, Z ) + α ( X ) h ( Y, Z ) = ( ∇ Y h )( X, Z ) + α ( Y ) h ( X, Z ) (Codazzi for h)3 ( ∇ X S )( Y ) − α ( X ) SY = ( ∇ Y S )( X ) − α ( Y ) SX (Codazzi for S)4 h ( X, SY ) − h ( SX, Y ) = dα ( X, Y ) (Ricci) Definition.
An affine connection ∇ is locally equiaffine if around each point x in M there is a parallel volume element,that is a non-vanishing n -from ω such that ∇ ω = . Definition.
An affine connection ∇ on M is called equiaffine connection if ∇ admits a parallel volume element ω on M , then we say ( ∇ , ω ) is an equiaffine structure. Definition.
Two torsion free locally equiaffine connections ∇ and ∇ ′ on a differentiable manifold M are said to beprojectively equivalent if there is a closed -from ρ such that ∇ ′ X Y = ∇ X Y + ρ ( X ) Y + ρ ( Y ) X, ∀ X, Y ∈ τ ( M ) . ∇ is called projectively flat if it is projectively equivalent to a flat connection. Definition.
Let R n +1 be the affine space with fixed point o . A hyper surface M embedded in R n +1 is said to becentro-affine hyper surface if the position vector −→ ox for each x in M is transversal to the tangent plane of M at x .Let D be the standard affine connection on R n +1 . Then ( D X Y ) x = ( ∇ YX ) x + h ( X, Y )( − x ) , ∀ x ∈ M , where ( ∇ YX ) x is tangent to M . ∇ is a connection on M , called the induced connection and h is called the affine fundamentalform. Note that the curvature tensor R and the Ricci tensor Ric of the induced connection on centro affine hypersurfaceare given by R ( X, Y ) Z = h ( Y, Z ) X − h ( X, Z ) Y and Ric ( Y, Z ) = ( n − h ( Y, Z ) respectively. Definition.
Let f : M −→ R n +1 be a hyper surface immersion, a transversal vector field ξ is called equiaffine if D X ξ is tangent to M for each X ∈ T x ( M ) . 3 efinition. Let f : M −→ R n +1 be a hyper surface immersion, a transversal vector field ξ is called non-degenerateif h is non-degenerate everywhere on M . Example.
Let o be the point of the affine space R n +1 chosen as origin, M be an n -dimensional manifold. Let f : M −→ R n +1 − { o } be an immersion such that the position vector −−−→ of ( x ) for x ∈ M is always transversal to f ( M ) . Now take ξ = −−−−→ of ( x ) , with respect to this ξ , f becomes an affine immersion.We have D X ξ = − f ∗ X so that α = and S = I . Proposition 1.2. [8] Let ( M , ∇ ) be n -manifold with projectively flat affine connection ∇ with symmetric Ricci tensor.Then ( M , ∇ ) can be locally immersed as a centro- affine hyper surface. Definition.
A transversal vector field is called Blaschke normal field if it satisfies(a) ( ∇ , η ) is an equiaffine structure, that is ∇ η = .(b) η coincides with the volume element ω h of non degenerated metric h , where η is the induced volume element.An affine immersion with Blaschke normal field is called Blaschke immersion. Definition.
A Blaschke hyper surface M is called an improper affine hyper shere if the shape operator S is identicallyzero.If S = λI where λ is a non zero constant, then M is called a proper affine hyper shere.
2. Statistical Manifold
The affine geometric aspects of statistical manifold are studied by Kurose [3, 4] and Uohashi,Ohara, Fujii [10].Families of probability measures is a subset of the infinite dimensional affine space of the positive measures up to ascale. Parametrized models are surfaces in this affine space. In this section we look at various cases in which the sta-tistical manifold is immersed as a hyper surface in R n +1 . We show that every centro-affine hypersurface is a statisticalmanifold realized in R n +1 , dually flat statistical manifold is locally immersed as a centro-affine hypersurface. Also if f : M −→ R n +1 be a Blaschke immersion, then ( M , ∇ , h ) is a statistical manifold. Definition.
Consider the family S of probability distributions on a sample space χ . Suppose each element of S canbe parametrized using n real valued variables { θ , θ , θ ...θ n } so that S = { p θ = p ( x, θ ) | θ = ( θ , θ , θ ...θ n ) ∈ U } , where U ⊆ R n and θ −→ p θ is injective. Such a family S is called n dimensional statistical model. There are someregularity conditions for statistical model which are important for the geometric study4 egularity conditions (a) U is an open subset of R n and for each x ∈ χ , the function θ −→ p ( x ; θ ) is of class C ∞ .(b) Let ℓ ( x ; θ ) = logp ( x ; θ ) and ∂ i = ∂∂θ i . For every fixed θ , n functions in x { ∂ i ℓ ( x ; θ ); i = 1 , , ..n } are linearlyindependent and are known as scores.(c) The order of integration and differentiation may be freely interchange.(d) Moment of scores exists upto necessary orders. Definition.
Let S = { p ( x, θ ) | θ ∈ U ⊆ R n } be a statistical model, the mapping φ : S −→ R n defined by φ ( p θ ) = θ allow us to consider φ = [ θ i ] as a coordinate system for S . Suppose we have a C ∞ diffeomorphism ψ from U onto ψ ( U ) ⊆ R n . If ρ = ψ ( θ ) then, S = { p ( x, ψ − ( ρ )) | ρ ∈ ψ ( U ) } . By considering parametrization which are C ∞ diffeomorphic to each other to be equivalent we may consider S as a smooth manifold, called the statistical manifold. Definition (Amri’s α - connection) . Let S = { p ( x, θ ) | θ ∈ U ⊆ R n } , be an n -dimensional statistical manifold g bethe Fisher information metric. Let α ∈ R . Define n functions Γ αij,k = E θ [( ∂ i ∂ j ℓ θ + 1 − α ∂ i ℓ θ ∂ j ℓ θ ) ∂ k ℓ θ ] Then we have an affine connection ∇ α on S defined by h∇ α∂ i ∂ j , ∂ k i = Γ αij,k , called Amri’s α - connections. Remark. ∇ α is flat iff ∇ − α is flat. If α = 1 then ∇ is called exponential connection or e - connection and we denoteit by ∇ or ∇ e , ∇ − called m -connection is also denoted by ∇ m . Definition (Lauritzen) . A statistical manifold is a pseudo-Riemannian manifold ( M , h ) with a symmetric covarianttensor ∇ h of order . Definition.
Let M be an n -dimensional manifold and h be a non-degenerate metric and ∇ be an affine connection.We define a connection ∇ by Xh ( Y, Z ) = h ( ∇ X Y, Z ) + h ( Y, ∇ X Z ) .This connection ∇ is called the conjugate connection. Note that the torsion tensor T and T of ∇ and ∇ arerespectively satisfy ( ∇ X h )( Y, Z ) + h ( Y, T ( X, Z )) = ( ∇ Y h )( X, Z ) + h ( Y, T ( X, Z )) (1)Then ∇ is torsion free if and only if ( ∇ , h ) satisfies the Codazzi’s equation ( ∇ X h )( Y, Z ) = ( ∇ Z h )( Y, X ) , ∀ X, Y, Z ∈ τ ( M ) . (2)Also the curvature tensor R and R of ∇ and ∇ are related by h ( R ( X, Y ) Z, U ) = − h ( Z, R ( X, Y ) U ) , (3)so R = if and only if R = . 5 efinition. Let M be an n -dimensional smooth manifold with affine connection ∇ . Then we have a connection incotangent bundle T ∗ ( M ) called the dual connection denoted by ∇ ∗ defined as ( ∇ ∗ X ω )( Y ) = X ( ω ( Y )) − ω ( ∇ X Y ) (4)where ω is a -from.Let M be a manifold with non-degenerate metric h . Then we can identify the tangent bundle T ( M ) and cotangentbundle T ∗ ( M ) in the following way. For each x ∈ M we have a linear isomorphism λ h : T ( M ) −→ T ∗ ( M ) definedby λ h ( X )( Y ) = h ( X, Y ) ∀ X, Y ∈ T x ( M ) (5) Remark.
Let M be a manifold with pseudo-Riemannian metric h and ∇ be an affine connection on M such that ∇ h is symmetric. The conjugate connection ∇ of ∇ relative to h and dual connection ∇ ∗ in T ∗ ( M ) are correspond toeach other by the isomorphism λ h . Thus for an n dimensional manifold ( M , ∇ , h ) with ∇ h symmetric we have a dualconnection ∇ ∗ on T ( M ) so that we have the duality structure ( M , ∇ , ∇ ∗ , h ) .On a statistical manifold we always have a duality structure ∇ ∗ . Also ( M , ∇ ) is flat if and only if ( M , ∇ ∗ ) is flat,in that case we call ( M , h, ∇ , ∇ ∗ ) is a dually flat space. Definition.
For a real number α two statistical manifolds ( M , ∇ , h ) and ( M , ˜ ∇ , ˜ h ) , are said to be ( α ) -conformallyequivalent if there exist a function φ on M such that ˜ h ( X, Y ) = e φ h ( X, Y ) , ˜ h ( ˜ ∇ X Y, Z ) = h ( ∇ X Y, Z ) − α dφ ( Z ) h ( X, Y ) + 1 − α { dφ ( X )˜ h ( Y, Z ) + dφ ( Y )˜ h ( X, Z ) } . A statistical manifold is said to be ( α ) -conformally flat if it is ( α ) -conformally equivalent to a flat connection. Note.
In [3, 4] Kurose discussed about affine immersion of statistical manifolds into the affine space.Let f : M −→ R n +1 , be an affine immersion with transversal vector field ξ and induced connection ∇ . From thedefinition of non-degenerate and equiaffine transversal vector fields, ( M , ∇ , h ) is a statistical manifold if and only if ξ is non-generate and equiaffine, in this case we say that ( M , ∇ , h ) realize in R n +1 and by preposition(9.1) in [11]statistical manifold ( M , ∇ , h ) is ( − -conformally flat if and only if ∇ is a projectively flat connection with symmetricRicci tensor. Also from [5] a simply connected statistical manifold ( M , ∇ , h ) can be realized in R n +1 if and only if ∇ ∗ is projectively flat. Proposition 2.1.
Every centro-affine hypersurface is a statistical manifold realized in R n +1 .Proof. Let R n +1 be an affine space with fixed point o , M be an n -dimensional manifold. Let f : M −→ R n +1 − { o } be an immersion such that the position vector −−−→ of ( x ) for x ∈ M is always transversal to f ( M ) . Now take ξ = −−−→ f ( x ) o ,6ith respect to this ξ , f becomes an affine immersion and we have D X ξ = − f ∗ ( X ) (negative sign comes because oforientation), so the transversal connection α = and the shape operator S = I . Then from the Codazzi for h , ∇ h issymmetric. So ( M , ∇ , h ) is a statistical manifold realized in R n +1 , where ∇ and h are the induced connection andaffine fundamental form respectively. Proposition 2.2.
Let ( M , h, ∇ , ∇ ∗ ) is a dually flat statistical manifold. Then both ( M , ∇ ) and ( M , ∇ ∗ ) can belocally immersed as a centro-affine hypersurface.Proof. Since ( M , ∇ ) is a flat manifold it is ( α ) -conformally flat for all α by taking φ = in the definition of ( α ) -conformally equivalent. So in particular ( M , ∇ ) is ( − -conformally flat. Then ∇ is projectively flat connection withsymmetric Ricci tensor. Then from the proposition (1 .
2) ( M , ∇ ) is locally immersed as a centro-affine hypersurface.Similarly we can prove for ( M , ∇ ∗ ) . Proposition 2.3.
Let f : M −→ R n +1 be a Blaschke immersion. Then ( M , ∇ , h ) is a statistical manifold, where ∇ is the induced connection and h is the induced affine fundamental form.Proof. Let f : M −→ R n +1 be a Blaschke immersion.Then by the definition of Blaschke immersion ∇ η = . Since ∇ X η = α ( X ) η for all vector field X , we have α = .Then from the Codazzi for h , ∇ h is symmetric and it follows that ( M , ∇ , h ) is a statistical manifold.
3. Exponential family
The set of probability measures is a subset of the affine space of positive measures up to scale. Parametrizedfamily of probability measures can be considered as a surface in this affine space of positive measures upto scale. Inparticular the geometry of exponential family is the simplest geometry, the affine geometry. Exponential families arecharacterized as the finite dimensional affine subspaces of the affine space of measures upto scale.In this section, based on the ± - flat structure of exponential family, we explore certain geometric propertiesof exponential family. We show that if all ∇ e auto parallel submanifold of a statistical manifold S are exponentialfamily, then S is an exponential family. We prove that submanifold of a statistical manifold which is an exponentialfamily is ∇ e -auto parallel submanifold. Also observed that exponential family is locally immersed as a centro-affinehypersurface. Definition.
Let S and M be manifolds with M ⊂ S . Let [ θ i ] = ( θ , θ , ...θ n ) and [ u a ] = ( u , u , ...u m ) be coordinatesystem for S and M respectively, where n = dim ( S ) and m = dim ( M ) . We call M a submanifold of S if thefollowing conditions hold. • The restriction θ i | M to M is a C ∞ - functions on M , i = 1 , , ...n . • Let B ia = ( ∂θ i | M ∂u a ) | p and B a = [ B a , B a ...B na ] ∈ R n . Then for each p ∈ M , { B , B , ....B m } are linearlyindependent, ( so, m ≤ n ) . 7 For any open subset W of M , there exists U an open subset of S , such that W = M ∩ U . Definition.
Let M be a submanifold of S and ∇ be an affine connection on S . M is said to be auto parallel withrespect to ∇ if ∇ X Y ∈ τ ( M ) for all X, Y ∈ τ ( M ) .1-dimensional auto parallel submanifolds are called geodesics. Remark.
A necessary and sufficient condition for M to be auto parallel is that ∇ ∂ a ∂ b ∈ τ ( M ) holds for all a, b , where ∂ a = ∂∂u a . Definition.
Let M be a submanifold of S . Let p ∈ M , then T p M ⊂ T p S . Now consider the projection map π p : T p M −→ T p S and π p ( V ) = V , ∀ V ∈ T p M . Let ∇ be a connection on S , then we define a connection ∇ π on M as ( ∇ πX Y ) p = π p ( ∇ X Y ) p , ∀ p ∈ M , ∀ X, Y ∈ τ ( M ) Define the second fundamental form or Embedding curvature as H ( X, Y ) = ∇ X Y − ∇ πX Y Remark.
Let S be an n -dimensional manifold and M be a submanifold of dimension m . For each p ∈ S , let { ( ∂ a ) p ; 1 ≤ a ≤ m } , be the basis for T p M and let { ( ∂ k ) p ; m + 1 ≤ k ≤ n } , be the basis for T p M ⊥ . Thenwe define m ( n − m ) functions { H abk } in the following way H abk = h H ( ∂ a , ∂ b ) , ∂ k i = h∇ ∂ a ∂ b , ∂ k i It follows that H = 0 iff H abk = 0 for ≤ a, b ≤ m and m + 1 ≤ k ≤ n . Also we have H ( X, Y ) = iff M is ∇ auto parallel submanifold of S . Definition.
Let X be a sample space and µ be a measure on X . The exponential family is the family of probabilitydistributions p ( x, θ ) expressed in the following form p ( x, θ ) = exp( r X i =1 θ i x i − K ( θ )) dµ, where { x , x , .....x r } are random variables, θ = { θ , θ , .....θ r } are parameters and K ( θ ) is the normalizer. Example.
Let p ( x, µ, σ ) = 1 √ πσ e − ( x − µ )22 σ be the normal family ,then take θ = − σ , θ = µσ , and K ( θ ) = log( − πθ )2 − ( θ ) θ Then we can easily see p ( x, µ, σ ) = p ( θ , θ ) = exp( x θ + xθ − K ( θ )) Note.
Exponential family is an affine space, it is also flat with respect to ∇ ± connection. ± connection need not be an exponential family. Example.
Let q be a smooth probability density function on R and q k be the k th iid extension. Then for Y = ( y , y , y , ...y k ) t (6) we have q k ( Y ) = q ( y ) q ( y ) q ( y ) , ...q ( y k ) (7) For a regular matrix A ∈ R k × k and a vector µ ∈ R k , we define a probability density function on R k by p ( A, µ, x ) = q k ( A − ( x − µ )) | det ( A ) | (8) Now define a statistical model S = { p ( A, µ, x ) | µ ∈ R k } (9) Now consider log ( p ( A, µ, x )) = k X i =1 log ( q ( A − ( x − µ ))) − log ( | det ( A ) | ) (10) Then clearly ∂log ( p ( A,µ,x )) ∂µ i is constant. So from the definition of Amari’s α connection Γ αij,k = , it implies that S is α -flat for all α , but in general it need not be an exponential family. Theorem 3.1. [6] Let S be an exponential family and M be a submanifold of S . Then M is exponential family iff M is auto parallel with respect to ∇ in S . Theorem 3.2.
Let S = { P ( x, θ ) | θ ∈ Θ } , be an n -dimensional statistical manifold with dually flat structure ( S, ∇ e , ∇ m ) . If all ∇ e auto parallel submanifolds of S are exponential family, then S is an exponential family.Proof. Let S = { P ( x, θ ) | θ ∈ Θ } , be an n -dimensional statistical manifold with dually flat structure ( S, ∇ e , ∇ m ) ,where g is the fisher information metric. Let θ = [ θ i ] and η = [ η j ] be the coordinate system of S with respect to ∇ e and ∇ m respectively. Now subdivide the range of index i = 1 , , ...n into indexing sets I = { i = 1 , , ...k } and II = { i = k + 1 , k + 2 , ...n } . Let M ( C II ) be the set of points whose coordinates [ θ i ] in II are fixed to constant C II = ( C iII ) for i = k + 1 , k + 2 , ..n . That is M ( C II ) = { p ∈ S | θ k +1 = C k +1 II , θ k +2 = C k +2 II , ...θ n = C nII } (11)Where C II ∈ R n − k , then clearly this is an affine space with respect to θ - coordinate system, which implies M ( C II ) is a ∇ e - auto parallel submanifold of S . Also if C II = C ′ II then M ( C II ) ∩ M ( C ′ II ) = φ , and S C II M ( C II ) = S .Now by our assumption M ( C II ) is an exponential family for all C II . If p ( x, θ ) ∈ S , then p ( x, θ ) ∈ M ( C II ) forsome constant C II , this implies that p ( x, θ ) = exp ( k X i =1 θ i x i − ψ β ( θ )) (12)9here ψ β ( θ ) defined on Θ β = { θ ∈ Θ | θ k +1 = C k +1 II , θ k +2 = C k +2 II , ...θ n = C nII } . Now define φ ( θ ) = ψ β ( θ ) if θ ∈ Θ β . Then we can write p ( x, θ ) = exp ( k X i =1 θ i x i − φ ( θ )) (13) = exp ( k X i =1 θ i x i + n X i = k +1 C iII x i − k X i =1 C iII x i − φ ( θ )) (14) = exp ( n X i =1 θ i x i + F ( x ) − φ ( θ )) (15)where F ( x ) = − P ki =1 C iII x i for p ( x, θ ) ∈ M ( C II ) , then S is an exponential family. Theorem 3.3.
Let S = { p ( x, θ ) | θ ∈ Θ } be a statistical manifold with ∇ e connection. Let M be a submanifold of S . If M is an exponential family then M is ∇ e auto parallel submanifold of S Proof.
Let S = { p ( x, θ ) | θ ∈ Θ } and M = { q ( x, u ) } be the submanifold of S , [ θ i ] be the coordinates on S and [ u a ] be the coordinates on M . Suppose M is an exponential family, then q ( x, u ) = p ( x, θ ( u )) = exp { n X a =1 u a G a ( x ) + D ( x ) − φ ( u ) } (16)we have, Γ ab,k = E ξ [( ∂ a ∂ a ℓ θ ) ∂ k ℓ θ ] , where ℓ θ = log ( p ( x, θ )) . Then ∂ a ∂ a ℓ θ = − ∂ φ∂u a ∂u b . Therefore we have, Γ ab,k = 0 which implies h∇ e∂ a ∂ b , ∂ k i = 0 , ∀ k . Hence H abk = 0 , which implies that M is a ∇ e auto parallelsubmanifold of S . Note.
From the proposition(2.2), exponential family with ± connection is locally immersed as a centro-affine hyper-surface Acknowledgements
The first author expresses his sincere gratitude to Indian institute of Space Science and technology, department ofspace, government of India for supporting his research work by providing fellowship and other necessary means.