Abstract
Barycentres of a discrete probability measure on a dually flat statistical manifold are introduced. They are shown to be unique and to behave as barycentres in Euclidean space. The estimation of these barycentres is studied. Potential applicative usefulness of informative barycentres include the problem of interpolating a statistical manifold valued map and the problem of model merging, which consists in merging several statistical models into a unique one. The results are illustrated on the exponential family, for which a projection theorem is proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amari, S. (1985).Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics, Vol. 28, Springer-Verlag, Berlin.
Amari, S. and Nagaoka, H. (2000).Methods of Information Geometry, AMS and Oxford University Press.
Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S. and Rao, C. (1987),Differential Geometry in Statistical Inference, Institute of Mathematical Statistics Lecture-Notes-Monograph Series, Vol. 10, Institute of Mathematical Statistics, Hayward, California.
Barndorff-Nielsen, O. (1988).Parametric Statistical Models and Likelihood, Lecture Notes in Statistics, Vol. 50, Springer-Verlag, New York.
Corcuera, J. and Kendall, W. (1999). Riemannian barycentres and geodesic convexity,Mathematical Proceedings of the Cambridge Philosophical Society,127 (2), 253–269.
Critchley, F., Mariott, P. and Salmon, M. (1994). Preferred point geometry and the local differential geometry of the kullback-leibler divergence,The Annals of Statistics,22(3), 1587–1602.
Csiszar, I. (1975). I-divergence geometry of probability distributions and minimization problems,The Annals of Probability,3, 146–158.
Dempster, A., Laird, N. and Rubin D. (1977). Maximum likelihood from incomplete data via the emalgorithm (with discussion),Journal of the Royal Statistical Society,39(1), 1–38.
Efron, B. (1975). Defining the curvature of a statistical problem (with applications to second order efficiency),The Annals of Statistics 3, 1189–1242.
Emery, M. and Meyer, P. (1989).Stochastic Calculus in Manifolds. With an Appendix by P.-A. Meyer, Universitext, Springer-Verlag, Berlin.
Everitt, B. and Hand, D. (1981).Finite Mixture Distributions Chapmann and Hall, New York.
Fujiwara, A. and Nagaoka, H. (1995), Quantum fisher metric and estimation for pure state models,Physics Letters,201A, 119–124.
Furman, W. and Lindsay, B. (1994). Testing for the number of components in a mixture of normal distributions using moment estimators,Computational Statistics and Data Analysis,17(5), 473–492.
Kass, R. and Vos, P. (1997).Geometrical Foundations of Asymptotix Inference, John Wiley, New York.
Kobayashi, S. and Nomizu, K. (1969).Foundations of Differential Geometry, Interscience Publishers John Wiley and sons Inc., New York, London, Sydney.
Komaki, F. (1996). On asymptotic properties of predictive distributions,Biometrika,83(2), 299–313.
McLachlan, G. and Peel, D. (2000).Frnite Mixture Models, Wiley, New York.
Murray, M. and Rice, J. (1993).Differential Geometry and Statistics, Monographs on Statistics and Applied Probability, Vol. 48, Chapman and Hall, London.
Oller, J. and Corcuera, J. (1995). Intrinsic analysis of statistical estimation,The Annals of Statistics,23(5), 1562–1581.
Pardo, J., Pardo, L. and Zografos, K. (2002) Minimum ϕ-divergence estimators with constraints in multinomial populations,Journal of Statistical Planning and Inference,104, 221–237.
Pelletier, B. (2002). Remote sensing of phytoplankton with neural networks,Proceedings of SPIE International Symposium on Remote Sensing, Vol. 4880, 158–169.
Pistone, G. and Sempi, C. (1995). An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one,The Annals of Statistics,23(5), 1543–1561.
Rao C. (1945). Information and the accuracy attainable in the esetimation of statistical parameters,Bulletin of the Calcutta Mathematical Society,37, 81–91.
Renyi, A. (1961) On measures of entropy and information,Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, 547–561, University of California Press, Berkeley.
Scott, D. and Szewczyk, W. (2001). From kernels to mixtures,Technometrics,43(3), 323–335.
Skovgaard, L. (1984). A riemannian geometry of the multivariate normal model,Scandinavian Journal of Statistics,11, 211–223.
Author information
Authors and Affiliations
About this article
Cite this article
Pelletier, B. Informative barycentres in statistics. Ann Inst Stat Math 57, 767–780 (2005). https://doi.org/10.1007/BF02915437
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02915437