Higher Order Expansions for Posterior Distributions Using Posterior Modes

The (arbitrary) higher order asymptotic expansion for posterior distributions of a single parameter is derived by using posterior modes, and its validity is shown. An asymptotic expansion for the Bayes risk with squared error loss of a posterior mode is derived.

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Published inJOURNAL OF THE JAPAN STATISTICAL SOCIETY Vol. 38; no. 3; pp. 415 - 429
Main Author Miyata, Yoichi
Format Journal Article
LanguageEnglish
Published Tokyo THE JAPAN STATISTICAL SOCIETY 01.01.2008
Japan Science and Technology Agency
Subjects
Asymptotic expansion
Bayesian inference
Laplace's method
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Abstract The (arbitrary) higher order asymptotic expansion for posterior distributions of a single parameter is derived by using posterior modes, and its validity is shown. An asymptotic expansion for the Bayes risk with squared error loss of a posterior mode is derived.
AbstractList The (arbitrary) higher order asymptotic expansion for posterior distributions of a single parameter is derived by using posterior modes, and its validity is shown. An asymptotic expansion for the Bayes risk with squared error loss of a posterior mode is derived.
ArticleNumber 415
Author Miyata, Yoichi
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References (4)Johnson, R. A. (1970). Asymptotic expansions associated with posterior distributions, Ann. Math. Statist., 41, 851–864.
(9)Petrov, V. V. (1975). Sums of Independent Random Variables, Akademie-Verlag, Berlin.
(8)Miyata, Y. (2004). Fully exponential Laplace approximations using asymptotic modes, J. Amer. Statist. Assoc., 99, 1037–1049.
(5)Johnson, R. A. and Ladalla, J. N. (1979). The large sample behavior of posterior distributions when sampling from multiparameter exponential family models and allied results, Sankhyā, Ser. B, 41, 196–215.
(10)Shimizu, R. (2001). Central Limit Theorem (in Japanese), Kyoiku Syuppan.
(3)Johnson, R. A. (1967). An asymptotic expansion for posterior distributions, Ann. Math. Statist., 38, 1899–1907.
(1)Crowder, M. (1988). Asymptotic Expansions of Posterior Expectations, Distributions and Densities for Stochastic Process, Ann. Inst. Statist. Math., 40(2), 297–309.
(7)Ladalla, J. N. (1988). A closed form approximation to the posterior distribution when sampling from multiparameter exponential family models, Sankhyā, Ser. A, 52, 192–217.
(2)Feller, W. (1971). An Introduction to Probability and its Applications, Vol. 2, Wiley, New York.
(6)Kass, R. E., Tierney, L. and Kadane, J. B. (1990). The validity of posterior expansions based on Laplace's method, in Bayesian and Likelihood Methods in Statistics and Econometrics (eds. S. Geisser, J. S. Hodges, S. J. Press, and A. Zellner), 473–488, Amsterdam NorthHolland.
(11)Singh, K. (1981). On the asymptotic accuracy of Efron's bootatrap, Ann. Statist., 9, 1187–1195.
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SubjectTerms Asymptotic expansion
Bayesian inference
Laplace's method
Title Higher Order Expansions for Posterior Distributions Using Posterior Modes
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