On the geometric structure ofsome statistical manifolds
aa r X i v : . [ m a t h . DG ] A p r On the geometric structure ofsome statistical manifolds
Mingao Yuan
Department of Statistics,North Dakota State University,Fargo, ND 58102, USA
E-mail: [email protected] 5, 2019
Abstract
In information geometry, one of the basic problem is to study the geomet-ric properties of statistical manifold. In this paper, we study the geometricstructure of the generalized normal distribution manifold and show that ithas constant α -Gaussian curvature. Then for any positive integer p , we con-struct a p -dimensional statistical manifold that is α -flat. Keywords : information geometry, statistical manifold, α -geometry,generalized normal distribution.
1. Introduction
Fisher information is an important quantity in probability and statistics.It measures the amount of information that an observable random variablecarries about the unknown parameters of the underlying distribution. Thewell-known Cramer-Rao theorem states that the lower bound of the varianceof any unbiased estimator is the inverse of the Fisher information. In asymp-totic theory, the maximum likelihood estimator converges in distribution toGaussian distribution with mean zero and variance the inverse of the Fisherinformation. In 1945, Rao noticed that the Fisher information defines a Rie-mannian metric on a statistical manifold([17]). Closely related to the Fisher1nformation is the statistical curvature defined on one-parameter distributionfamily by Bradley Efron([11]). It controls how much the variance of the max-imum likelihood estimator exceeds the Cramer-Rao lower bound([11]). LaterMadsen extended the result of Efron to the multi-parameter case([14]). It’swell-known that differential geometry is an important field in mathematics.The famous Einstein’s relativity theory depends on Riemannian geometryand recently some researchers are interested in extending the relativity theoryby using the more general Riemann-Finsler geometry. See [4, 7, 8, 9, 18, 19]for some references. In 1982, Amari provided a differential geometrical frame-work for analyzing statistical problmes related to mult-parameter families ofdistribution and introduced the α -geometry on statistical manifold([1]). The α -geometry measures the second-order information loss and second-order ef-ficiency of an estimator([1]). Since then, many researchers studied the ge-ometry of the statistical manifold([1][2][11][12]). Amari, Arwini and Dodsonstudied the α -geometry of Gaussian, Gamma, Mckey bivariate gamma andthe Freund bivariate exponential manifold([2][3]). Recently, the α -geometryof Weibull, inverse gamma distribution, t-distribution and generalized expo-nential distribution manifold are investigated([5][6][13]). One interesting factis that the Gaussian manifold and the Weibull manifold have negative con-stant Gaussian curvature([3][5]) and several of the submanifolds of the Fre-und bivariate exponential manifold are α -flat([3]). The statistical manifoldwith negative constant α -curvature will share similar statistical properties asGaussian manifold and Weibull manifold([1][11]). Especially, the MLE forsome parameter in α -flat statistical manifold has no second order informa-tion loss([1][11]). Then one both statistically and geometrically interestingquestion is whether we have other statistical manifolds that have constantGaussian curvature or α -Gaussian curvature. In this paper, we firstly showthat the generalized Gaussian statistical manifold has constant α -Gaussiancurvature. Then for any positive integer p , we construct a p -dimensionalstatistical manifold that is α -flat.The generalized Gaussian distribution is a generalization of the normaland Laplace distributions. It has received widespread applications in manyapplied areas([15][16]). The generalized Gaussian distribution manifold isdefined as M = (cid:26) f ( x ; µ, σ, β ) | f ( x ; µ, σ, β ) = β σ Γ(1 /β ) e − | x − µ | βσβ , x, µ ∈ R , σ, β > (cid:27) , where µ , σ , β are called the location, scale and shape parameters respectively2nd Γ( x ) is the gamma function. Clearly, this fimily includes the Gaussiandistribution when β = 2 and the Laplace distribution if β = 1. Note that if β is odd, the manifold is not smooth. Hence we only consider the case when β is a known even number. Theorem 1.
Let β be a given even number. Then the Riemannian metricon the generalized Gaussian statistical manifold M is ( g ij ) = (cid:20) σ c σ c (cid:21) , (1) where c = Γ(1 − β ) β ( β − β ) , c = β. The α -curvature tensor is given by R ( α )1212 = − (1 − α ) β ( β − − β + (1 − α )( β − β − β ) σ Γ( β ) , (2) and the α -Gaussian curvature is constant and given by K ( α ) = − (1 − α ) (cid:0) − β + (1 − α )( β − (cid:1) β . (3) Then the α -curvature tensor vanishes if and only if α = 1 or β − . Note that when α = 0, the K (0) is the Gaussian curvature of the Rie-mannian metric. In this case, K (0) = − β . If β = 2, then the manifold isjust the univariate Gaussian manifold. Plugging β = 2 into the formula for K (0) , we get K (0) = − , which is the same as that in ([3]). When β = 4, K (0) = − , and K (0) = − for β = 6. Then by using Theorem 1, we can geta lot of non-Gaussian statistical manifolds with constant Gaussian curvaturedifferent from that of the Gaussian manifold and the Weibull distribution.Next, we define another interesting non-Gaussian statistical manifold. LetΩ p = { x = ( x , . . . , x p ) ∈ R p | Q pi =1 x i > } and R p + = { x = ( x , . . . , x p ) ∈ R p | x i > , i = 1 , , . . . , p. } , we define a p -dimensional statistical manifold M = (cid:26) f ( x ; λ ) | f ( x ; λ ) = 2 p Y i =1 √ λ i √ π e − λix i , x ∈ Ω p , λ ∈ R p + (cid:27) . p = 2,we have f ( x , x ) = 2 √ λ √ π e − λ x √ λ √ π e − λ x I [ x x > , and the marginal distribution f X ( x ) = Z + ∞−∞ √ λ √ π e − λ x √ λ √ π e − λ x I [ x x > dx = Z −∞ √ λ √ π e − λ x √ λ √ π e − λ x I [ x x > dx + Z + ∞ √ λ √ π e − λ x √ λ √ π e − λ x I [ x x > dx = √ λ √ π e − λ x I [ x <
0] + √ λ √ π e − λ x I [ x > √ λ √ π e − λ x , ( x ∈ R ) , where I is the indicator function. Obviously f is not a Gaussian densitybut f X is the density of the Gaussian distribution with mean zero and vari-ance λ . Similarly, one can show that the another marginal distribution isGaussian distribution with men zero and variance λ .For this non-Gaussian manifold M , we have Theorem 2.
For any positive integer p , the p -dimensional statistical mani-fold M is α -flat. By this theorem, there exists α -flat statistical manifold with any dimen-sion.
2. Geometry of statistical manifold
Let M = { p ( x ; θ ) | θ ∈ Θ ⊂ R p } be a statistical manifold, l = log p ( x ; θ )and ∂ i = ∂∂θ i . The Riemannian metric on M is defined by g ij ( θ )] = − E (cid:2) ∂ i ∂ j l (cid:3) . kij = g kl (cid:26) ∂g li ∂θ j + ∂g lj ∂θ i − ∂g ij ∂θ l (cid:27) , and Γ ijk = Γ mij g mk . The α -connection is defined byΓ ( α ) ijk = E (cid:20)(cid:18) ∂ i ∂ j l + 1 − α ∂ i l∂ j l (cid:19) ∂ k l (cid:21) . Let T ijk = E (cid:2) ∂ i l∂ j l∂ k l (cid:3) and Γ (1) ijk = E [ ∂ i ∂ j l∂ k l ]. Then we haveΓ ( α ) ijk = Γ (1) ijk + 1 − α T ijk . Let Γ ( α ) kij = g km Γ ( α ) ijm . The α -curvature tensor is R ( α ) lihj = ∂ i Γ ( α ) lhj − ∂ h Γ ( α ) lij + X m Γ ( α ) lim Γ ( α ) mhj − X m Γ ( α ) lhm Γ ( α ) mij , and R ( α ) ihjk = X l g lk R ( α ) lihj . A statistical manifold is said to be α - flat if its α -curvature vanishes. For p = 2, the α -Gaussian curvature is defined as K ( α ) = R ( α )1212 det ( g ij ) . Note that the 0-geometry corresponds to the geometry of the Riemannianmetric.
3. Proof of the theorems
For distribution in M and β = 1 ,
2, we need to make a transformation ofthe parameter space so that the distribution can written as a regular expo-nential distribution. It is not easy to find such transformation for every β .So we work with the original parameter space without transformation. The5omputation is plausible. Proof of Theorem 1:
The log-likelihood function of the generalizednormal distribution is l = log f ( x ; µ, σ, β ) = log β − log (cid:18) β ) (cid:19) − log σ − ( x − µ ) β σ β . Then direct computation yields the first and second partial derivatives below ∂l∂µ = βσ β ( x − µ ) β − ,∂l∂σ = − σ + βσ β +1 ( x − µ ) β ,∂ l∂µ = − β ( β − σ β ( x − µ ) β − ,∂ l∂µ∂σ = − β σ β +1 ( x − µ ) β − ,∂ l∂σ = 1 σ − β ( β + 1) σ β +2 ( x − µ ) β . In terms of gamma function, we have the k -th moment E [( x − µ ) k ] = ( , k : odd, β : even ; Γ( k +1 β )Γ( β ) σ k , k, β : even. Notice that we assume β is even, then β − g = − E (cid:20) ∂ l∂µ (cid:21) = β ( β − σ β E (cid:20) ( x − µ ) β − (cid:21) = Γ(1 − β ) β ( β − β ) 1 σ ,g = − E (cid:20) ∂ l∂σ (cid:21) = − σ + β ( β + 1) σ β +2 E (cid:20) ( x − µ ) β (cid:21) = − σ + β ( β + 1) σ β +2 Γ( β +1 β )Γ( β ) σ β = βσ ,g = g = − E (cid:20) ∂ l∂µ∂σ (cid:21) = β σ β +1 E (cid:20) ( x − µ ) β − (cid:21) = 0 , T ijk below T = E (cid:20) ∂l∂µ ∂l∂µ ∂l∂σ (cid:21) = − β σ β +1 E (cid:20) ( x − µ ) β − (cid:21) + β σ β +1 E (cid:20) ( x − µ ) β − (cid:21) = 1 σ Γ( β − β ) β − Γ( β − β ) β Γ( β ) ,T = E (cid:20) ∂l∂σ ∂l∂σ ∂l∂σ (cid:21) = E (cid:20)(cid:18) − σ + βσ β +1 ( x − µ ) β (cid:19) (cid:21) = − σ + 3 βσ β +3 E (cid:20)(cid:18) x − µ (cid:19) β (cid:21) − β σ β +3 E (cid:20)(cid:18) x − µ (cid:19) β (cid:21) + β σ β +3 E (cid:20)(cid:18) x − µ (cid:19) β (cid:21) = 1 σ (cid:18) − β Γ( β +1 β )Γ( β ) − β Γ( β +1 β )Γ( β ) + β Γ( β +1 β )Γ( β ) (cid:19) = 2 β σ ,T = T = T ,T = T = T = T = 0 . The 1-connection coefficients areΓ (1)112 = E (cid:20) ∂ l∂µ ∂l∂σ (cid:21) = β ( β − σ β +1 E (cid:20) ( x − µ ) β − (cid:21) − β ( β − σ β +1 E (cid:20) ( x − µ ) β − (cid:21) , = 1 σ Γ( β − β ) β ( β − − Γ( β − β ) β ( β − β ) , Γ (1)121 = E (cid:20) ∂ l∂µ∂σ ∂l∂µ (cid:21) = − β σ β +1 E (cid:20) ( x − µ ) β − (cid:21) = − σ Γ( β − β ) β Γ( β ) , (1)222 = E (cid:20) ∂ l∂σ ∂l∂σ (cid:21) = − σ + βσ β +3 E (cid:20) ( x − µ ) β (cid:21) + β ( β + 1) σ β +3 E (cid:20) ( x − µ ) β (cid:21) − β ( β + 1) σ β +3 E (cid:20) ( x − µ ) β (cid:21) = 1 σ (cid:18) − β Γ( β +1 β )Γ( β ) + β ( β + 1)Γ( β +1 β )Γ( β ) − β ( β + 1)Γ( β +1 β )Γ( β ) (cid:19) = − β ( β − σ , Γ (1)211 = Γ (1)121 , Γ (1)111 = Γ (1)122 = Γ (1)212 = Γ (1)221 = 0 . The α -connection is just a linear combination of the 1-connection and T .Hence, the α -connection coefficients areΓ ( α )112 = Γ (1)112 + 1 − α T = (cid:18) c (1)112 + 1 − α c (cid:19) σ , Γ ( α )121 = Γ (1)121 + 1 − α T = (cid:18) c (1)121 + 1 − α c (cid:19) σ , Γ ( α )222 = Γ (1)222 + 1 − α T = (cid:18) c (1)222 + 1 − α c (cid:19) σ , Γ ( α )211 = Γ ( α )121 , Γ ( α )111 = Γ ( α )122 = Γ ( α )212 = Γ ( α )221 = 0 . To compute the α -curvature, we need the α -connection coefficients in aanother form. 8 ( α )211 = g Γ ( α )112 = 1 σ c (cid:18) c (1)112 + 1 − α c (cid:19) , Γ ( α )121 = g Γ ( α )211 = 1 σ c (cid:18) c (1)121 + 1 − α c (cid:19) , Γ ( α )112 = Γ ( α )121 , Γ ( α )221 = Γ ( α )212 = 0 . By definition, the α -curvature is R ( α )1212 = − h(cid:18) ∂∂σ Γ ( α )211 − ∂∂µ Γ ( α )221 (cid:19) g + Γ ( α )222 Γ ( α )211 − Γ ( α )112 Γ ( α )121 i = − C + C − C σ , (4)where the constants dependent on α and β are defined below c = Γ(1 − β ) β ( β − β ) ,c = β,C = − (cid:18) c (1)112 + 1 − α c (cid:19) ,C = 1 c (cid:18) c (1)222 + 1 − α c (cid:19)(cid:18) c (1)112 + 1 − α c (cid:19) ,C = 1 c (cid:18) c (1)112 + 1 − α c (cid:19)(cid:18) c (1)121 + 1 − α c (cid:19) , = Γ( β − β ) β − Γ( β − β ) β Γ( β ) ,c = c ,c = 2 β ,c (1)112 = Γ( β − β ) β ( β − − Γ( β − β ) β ( β − β ) ,c (1)121 = − Γ( β − β ) β Γ( β ) ,c (1)222 = − β ( β − . Then we can easily get the α -Gaussian curvature below K ( α ) = R ( α )1212 det ( g ij ) = − C + C − C c c . (5)Next, we simplify (4) and (5), as pointed out by Professor Esmaeil Peyghan.Note that C + C − C = (cid:16) c (1)112 + 1 − α c (cid:17)(cid:16) − c (1)222 c + 1 − α c c − c (1)121 c − − α c c (cid:17) . The first product factor can be calculated as c (1)112 + 1 − α c = Γ( β − β ) β ( β − − β + (1 − α )( β − β ) , where we used the fact that Γ( β − β ) = (2 β − β − β Γ( β − β ) and Γ( β − β ) = β − β Γ( β − β ), since Γ(1 + x ) = x Γ( x ). For the second product factor, we have − c (1)222 c − c (1)121 c = − β + β Γ( β − β )( β − β − β ) = − β + β β − β Γ( β − β )( β − β − β ) = 0 , − α c c − − α c c = 1 − α (cid:16) β − β Γ( β − β ) − β Γ( β − β )( β − β − β ) (cid:17) = 1 − α (cid:16) β − (2 β − β − β − β ) − ( β − β − β )( β − β − β ) (cid:17) = 1 − α. R ( α )1212 = − (1 − α ) Γ( β − β ) β ( β − − β + (1 − α )( β − σ Γ( β ) , which is (2). In this case, the α -Gaussian curvature is K ( α ) = − (1 − α ) Γ( β − β ) β ( β − − β + (1 − α )( β − β ) Γ( β )Γ( β − β ) β ( β − − (1 − α )(2 − β + (1 − α )( β − β , which is (3).For distribution in M , we can easily write it as a regular exponentialdistribution. Then we can work on the potential function to get the α -curvature([3]). Proof of Theorem 2:
We rewrite the distribution in M as f ( x ; λ ) = e P pi =1 log( λ i ) − P pi =1 λ i x i +log 2 − log √ π = e P pi =1 log( − θ i )+ P pi =1 θ i x i + p log 2 − log √ π , where θ i = − λ i . This is one member of the exponential family with( θ , . . . , θ p ) the natural coordinates and the potential function ψ ( θ ) = − p X i =1 log( − θ i ) . For exponential family, the Fisher information is just the second derivativeof the potential function([3]): g ij = ∂ ψ∂θ i ∂θ j = −
12 1 θ i θ j δ ij , where δ ii = 1 for i = 1 , . . . , p and δ ij = 0 for i = j . The third derivative ofthe potential function will give us the α -connectionΓ ( α ) ijk = 1 − α ∂ ψ∂θ i ∂θ j ∂θ k = − − α θ i θ j θ k δ ijk , δ iii = 1 for i = 1 , . . . , p and δ ijk = 0 for unequal i, j, k .Then Γ ( α ) kij = g kl Γ ( α ) ijl = − − α ( θ i θ j θ k ) δ ijk . Note that Γ ( α ) kij and Γ ( α ) ijk vanish when i, j, k are unequal. Hence the α -curvature also vanish, that is, R ( α ) hijk = 0 , which completes the proof. Acknowledgement
Sincere thanks to Professor Esmaeil Peyghan for the valuable commentsthat significantly improve this manuscript.
ReferencesReferences [1] Amari, S. I. (1982). Differential geometry of curved exponential families-curvatures and information loss.
The Annals of Statistics , , 357-385.[2] Amari, S. I. (1985). Differential geometrical methods in statistics.Springer Lecture Notes in Statistics, , Springer-Verlag, Berlin.[3] Arwini, K.A. and Dodson, C.T.J. (2008). Information Geometry-NearRandomness and Near Independence. Springer-Verlag Berlin Heidelberg .[4] Boothby, W.(2002).
An Introduction to Differentiable Manifolds and Rie-mannian Geometry ,Academic Press.[5] Cao, L.M., Sun H.F. and Wang X.J.(2008). The geometric structure ofthe Weibull distribution manifold and the generalized exponential distri-bution manifold.
Tamkang journal of mathematics , , 45-51.[6] Cho, B.S. and Jung, S.Y. (2010). A note on the geometric structure of thet-distribution. Journal of the Korean Data Information Science Society , , 575-580. 127] Chern,S.S. and Shen,Z.(2005). Riemann-Finsler Geometry, World Scien-tific Publishing Company.[8] Cheng, X.Y. and Yuan, M.(2014). On Randers metrics of isotropic scalarcurvature, Publicationes Mathematicae-Debrecen , :63-74.[9] Cheng,X.Y., Zhang, T. and Yuan, M.(2014). On dually flat and confor-mally flat ( α , β )-metrics, J. of Math. (PRC) , .[10] Dutta S. and Genton M.(2014). A non-Gaussian multivariate distribu-tion with all lower-dimensional Gaussians and related families. Journalof Multivariate Analysis , , 82-93.[11] Efron, B. (1975). Defining the curvature of a statistical problem. Annalsof Statistics , , 1109-1242.[12] Kass, R. E. (1989). The geometry of asymptotic inference. StatisticalScience , , 188- 219.[13] Li, T.Z., Peng, L.Y., and Sun, H.F.(2008). The geometric structureof the inverse gamma distribution. Beitrge zur Algebra und Geometrie , , 217-225.[14] Madsen, L. T. (1979). The geometry of statistical model-a generalizationof curvature. Research. Report. , Statist. Res. Unit., Danish MedicalRes. Council.[15] Nadarajah, S. (2005). A generalized normal distribution.
Journal of Ap-plied Statistics , , 685-694.[16] Pogany, T.K. and Nadarajah, S. (2010). On the characteristic functionof the generalized normal distribution. Comptes Rendus Mathematique , , 203-206.[17] Radhakrishna Rao, C. (1945). Information and accuracy attainable inthe estimation of statistical parameters. Bulletin of the Calcutta Mathe-matical Society ,
37 (3) , 81-91.[18] Shen, Z. and Yuan, M.(2016). Conformal vector fields on some Finslermanifolds.
Science China Mathematics , : 107-114.1319] Yuan, M. and Cheng, X.Y.(2015). On conformally flat ( α , β )-metricswith special curvature properties. Acta Mathematica Sinica, English Se-ries ,31