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SJK [email protected] Tanaka Fuyuhiko

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Page 1: Seminar 130107 ver2_public_embed

SJK������� [email protected]

Tanaka Fuyuhiko

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ª (1/2)ijvw�°Â+,ÍS. Amari:Differential geometrical methods in statistics. Springer-Verlag, 1985.S. Amari and H. Nagaoka,: Methods of Information Geometry. AMS, Oxford, 2000.

+/89%Q«�Ffª @+&NdC¬÷`¬�defª �×�ªC�­A/ % � ª @+ N C¬ ¬� ª ×�ªC�­A

J. Aitchison: Goodness of prediction fit. Biometrika, 62 (1975), 547-554.S. Amari: Differential geometry of a parametric family of invertible linear systems - Riemannian metric, dual

affine connections, and divergence. Math. Syst. Theory, 20 (1987), 53-82. K A t L' l h i l i i à b i i é ti I J l fK. Aomoto: L'analyse harmonique sur les espaces riemanniens, à courbure riemannienne négative I. Journal of

the Faculty of Science, University of Tokyo, 13 (1966), 85-105. J. Berger and R. Y. Yang, Noninformative priors and Bayesian testing for the AR(1) model. Econometric Theory,

10 (1994), 461-482.( ),J. M. Bernardo: Reference posterior distributions for Bayesian inference. J. R. Statist. Soc. B, 41, (1979), 113-147.J. M. Corcuera and F. Giummolé : A Generalized Bayes Rule for Prediction. Scand. J. Statist., 26, Issue 2,

(1999), 265-279. B A Frigyik S Srivastava and M R Gupta: Functional Bregman divergence and Bayesian estimation ofB.A.Frigyik, S.Srivastava, and M.R.Gupta: Functional Bregman divergence and Bayesian estimation of

distributions. IEEE Trans. Info. Theory, 54 no. 11 (2008), 5130-5139J. A. Hartigan: The Maximum Likelihood Prior. Ann. Statist., 26 no.6 (1998), 2083-2103. F. Komaki: Shrinkage priors for Bayesian prediction. Ann. Statist., 34 (2006), 808-819. g p y p ( )F. Komaki: A shrinkage predictive distribution for multivariate normal observables. Biometrika, 88 (2001), 859-

864.

Page 93: Seminar 130107 ver2_public_embed

ª (2/2)S. Lauritzen: Statistical manifolds. In Differential Geometry in Statistical Inference, IMS Lecture Notes:

Monograph Series 10:Institute of Mathematical Statistics, Hayward, California, (1987) 163-216.y ( )H. Matsuzoe, J. Takeuchi, and S. Amari, Equiaffine structures on statistical manifolds and Bayesian statistics.

Differential Geom. Appl., 24 (2006), 567-578.P.C.B. Phillips: To criticize the critics: an objective Bayesian analysis of stochastic trends. J. Appl. Econ. 6

(1991) 333 364(1991), 333-364.J. Takeuchi and S. Amari, �-parallel prior and its properties. IEEE. Trans. Info. Theory, 51, no.3 (2005), 1011-

1023.F. Tanaka, Superharmonic priors for autoregressive models. Mathematical Engineering Technical , p p g g g

Reports, 2009-18, (2009) 1-20.F. Tanaka and F. Komaki:Asymptotic expansion of the risk difference of the Bayesian spectral density in the autoregressive

moving average model, Sankhya Series A, Indian Statistical Institute, Vol.73-A (2011), pp. 162®184.F. Tanaka: Curvature form on statistical model manifolds and its application to Bayesian analysis, Journal of Statistics Applications and Probability, Natural Sciences Publishing, Vol.1 (2012), 35-43.F. Tanaka: Noninformative prior in the quantum statistical model of pure states. Phys. Rev. A, 85

(2012): 062305.