Kolassa, John E.; Kuffner, Todd A. On the validity of the formal Edgeworth expansion for posterior densities. (English) Zbl 1458.62065 Ann. Stat. 48, No. 4, 1940-1958 (2020). The authors establish validity of a formal Edgeworth expansion for the posterior density in a Bayesian setting. Under some regularity conditions (including, for example, differentiability, continuity and measurability conditions), they establish the order of the error in approximating the posterior density of \((\theta-\theta_0)/\sigma\) by an Edgeworth expansion involving Hermite polynomials and the cumulants of this statistic, where \(\theta\) is the relevant parameter, and \(\theta_0\) and \(\sigma\) are its posterior mean and standard deviation, respectively. A similar result is also established for the posterior distribution function. These results require estimates of the asymptotic order of posterior cumulants which the authors establish. The paper concludes with a comparison of the authors’ results with other, existing expansions (including those in which \(\theta\) is normalised using the MLE and observed information, for example) and numerical illustrations of the present results. Reviewer: Fraser Daly (Edinburgh) Cited in 3 Documents MSC: 62E20 Asymptotic distribution theory in statistics 62F15 Bayesian inference 60F05 Central limit and other weak theorems Keywords:Edgeworth expansion; higher-order asymptotics; posterior; cumulant expansion PDFBibTeX XMLCite \textit{J. E. Kolassa} and \textit{T. A. Kuffner}, Ann. Stat. 48, No. 4, 1940--1958 (2020; Zbl 1458.62065) Full Text: DOI arXiv Euclid References: [1] Barndorff-Nielsen, O. and Cox, D. R. (1979). Edgeworth and saddle-point approximations with statistical applications. J. Roy. Statist. Soc. Ser. B 41 279-312. Zentralblatt MATH: 0424.62010 Digital Object Identifier: doi:10.1111/j.2517-6161.1979.tb01085.x · Zbl 0424.62010 [2] Barndorff-Nielsen, O. E. and Cox, D. R. (1989). Asymptotic Techniques for Use in Statistics. Monographs on Statistics and Applied Probability. CRC Press, London. Zentralblatt MATH: 0672.62024 · Zbl 0672.62024 [3] Bertail, P. and Lo, A. Y. (2002). On Johnson’s asymptotic expansion for a posterior distribution. Technical Report No. 2002-39, Institut National de la Statistique et des Estudes Economiques, Série des Documents de Travail du CREST (Centre de Recherche en Economie et Statistique). [4] Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451. Zentralblatt MATH: 0396.62010 Digital Object Identifier: doi:10.1214/aos/1176344134 Project Euclid: euclid.aos/1176344134 · Zbl 0396.62010 [5] Bhattacharya, R. N. and Rao, R. R. (2010). Normal Approximation and Asymptotic Expansions. Classics in Applied Mathematics 64. SIAM, Philadelphia, PA. Zentralblatt MATH: 1222.41002 · Zbl 1222.41002 [6] Bickel, P. J. and Ghosh, J. K. (1990). A decomposition for the likelihood ratio statistic and the Bartlett correction—A Bayesian argument. Ann. Statist. 18 1070-1090. Zentralblatt MATH: 0727.62035 Digital Object Identifier: doi:10.1214/aos/1176347740 Project Euclid: euclid.aos/1176347740 · Zbl 0727.62035 [7] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. Wiley, New York. Zentralblatt MATH: 0822.60002 · Zbl 0822.60002 [8] Brillinger, D. R. (1969). The calculation of cumulants via conditioning. Ann. Inst. Statist. Math. 21 215-218. Zentralblatt MATH: 0181.46103 Digital Object Identifier: doi:10.1007/BF02532246 · Zbl 0181.46103 [9] Chang, I. H. and Mukerjee, R. (2006). Probability matching property of adjusted likelihoods. Statist. Probab. Lett. 76 838-842. Zentralblatt MATH: 1089.62024 Digital Object Identifier: doi:10.1016/j.spl.2005.10.015 · Zbl 1089.62024 [10] DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. Springer, New York. Zentralblatt MATH: 1154.62001 · Zbl 1154.62001 [11] Datta, G. S. and Mukerjee, R. (2004). Probability Matching Priors: Higher Order Asymptotics. Lecture Notes in Statistics 178. Springer, New York. Zentralblatt MATH: 1044.62031 · Zbl 1044.62031 [12] Davis, A. W. (1976). Statistical distributions in univariate and multivariate Edgeworth populations. Biometrika 63 661-670. Zentralblatt MATH: 0348.62008 Digital Object Identifier: doi:10.1093/biomet/63.3.661 · Zbl 0348.62008 [13] Davison, A. C. (1986). Approximate predictive likelihood. Biometrika 73 323-332. Zentralblatt MATH: 0595.62032 Digital Object Identifier: doi:10.1093/biomet/73.2.323 · Zbl 0595.62032 [14] DiCiccio, T. J. and Stern, S. E. (1993). On Bartlett adjustments for approximate Bayesian inference. Biometrika 80 731-740. Zentralblatt MATH: 0800.62157 Digital Object Identifier: doi:10.1093/biomet/80.4.731 · Zbl 0800.62157 [15] DiCiccio, T. J. and Stern, S. E. (1994). Frequentist and Bayesian Bartlett correction of test statistics based on adjusted profile likelihoods. J. Roy. Statist. Soc. Ser. B 56 397-408. Zentralblatt MATH: 0796.62016 Digital Object Identifier: doi:10.1111/j.2517-6161.1994.tb01989.x · Zbl 0796.62016 [16] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York. Zentralblatt MATH: 0219.60003 · Zbl 0219.60003 [17] Ghosh, J. K. (1994). Higher-Order Asymptotics. IMS, Hayward. [18] Ghosh, J. K., Delampady, M. and Samanta, T. (2006). An Introduction to Bayesian Analysis: Theory and Methods. Springer Texts in Statistics. Springer, New York. Zentralblatt MATH: 1135.62002 · Zbl 1135.62002 [19] Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer Series in Statistics. Springer, New York. [20] Ghosh, J. K., Sinha, B. K. and Joshi, S. N. (1982). Expansions for posterior probability and integrated Bayes risk. In Statistical Decision Theory and Related Topics, III, Vol. 1 (West Lafayette, Ind., 1981) (S. Gupta and J. Berger, eds.) 403-456. Academic Press, New York. [21] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer Series in Statistics. Springer, New York. Zentralblatt MATH: 0829.62021 · Zbl 0829.62021 [22] Hartigan, J. A. (1965). The asymptotically unbiased prior distribution. Ann. Math. Stat. 36 1137-1152. Zentralblatt MATH: 0133.42106 Digital Object Identifier: doi:10.1214/aoms/1177699988 Project Euclid: euclid.aoms/1177699988 · Zbl 0133.42106 [23] James, G. S. (1955). Cumulants of a transformed variate. Biometrika 42 529-531. Zentralblatt MATH: 0066.11902 Digital Object Identifier: doi:10.1093/biomet/42.3-4.529 · Zbl 0066.11902 [24] James, G. S. (1958). On moments and cumulants of systems of statistics. Sankhyā 20 1-30. Zentralblatt MATH: 0083.14902 · Zbl 0083.14902 [25] James, G. S. and Mayne, A. J. (1962). Cumulants of functions of random variables. Sankhyā, Ser. A 24 47-54. Zentralblatt MATH: 0105.12403 · Zbl 0105.12403 [26] Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Statistical Science Series 16. Oxford University Press, New York. [27] Johnson, R. A. (1967). An asymptotic expansion for posterior distributions. Ann. Math. Stat. 38 1899-1906. Zentralblatt MATH: 0157.46802 Digital Object Identifier: doi:10.1214/aoms/1177698624 Project Euclid: euclid.aoms/1177698624 · Zbl 0157.46802 [28] Johnson, R. A. (1970). Asymptotic expansions associated with posterior distributions. Ann. Math. Stat. 41 851-864. Zentralblatt MATH: 0204.53002 Digital Object Identifier: doi:10.1214/aoms/1177696963 Project Euclid: euclid.aoms/1177696963 · Zbl 0204.53002 [29] Kass, R. E., Tierney, L. and Kadane, J. P. (1990). The validity of posterior expansions based on Laplace’s method. In Bayesian and Likelihood Methods in Statistics and Econometrics (S. Geisser, J. S. Hodges, S. J. Press and A. Zellner, eds.) 473-488. Elsevier Science Publishers B.V., Amsterdam. Zentralblatt MATH: 0734.62034 · Zbl 0734.62034 [30] Kharroubi, S. A. and Sweeting, T. J. (2016). Exponential tilting in Bayesian asymptotics. Biometrika 103 337-349. Zentralblatt MATH: 07072115 Digital Object Identifier: doi:10.1093/biomet/asw018 · Zbl 1499.62408 [31] Kolassa, J. E. (2006). Series Approximation Methods in Statistics, 3rd ed. Lecture Notes in Statistics 88. Springer, New York. Zentralblatt MATH: 1090.62015 · Zbl 1090.62015 [32] Leonov, V. P. and Sirjaev, A. N. (1959). On a method of semi-invariants. Theory Probab. Appl. 4 319-329. [33] Lindley, D. V. (1961). The use of prior probability distributions in statistical inference and decisions. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 453-468. Univ. California Press, Berkeley, CA. [34] Lindley, D. V. (1980). Approximate Bayesian methods. Trab. Estad. Investig. Oper. 31 223-237. Zentralblatt MATH: 0458.62002 · Zbl 0458.62002 [35] McCullagh, P. (1987). Tensor Methods in Statistics. Monographs on Statistics and Applied Probability. CRC Press, London. Zentralblatt MATH: 0732.62003 · Zbl 0732.62003 [36] Mykland, P. A. (1999). Bartlett identities and large deviations in likelihood theory. Ann. Statist. 27 1105-1117. Zentralblatt MATH: 0951.62014 Digital Object Identifier: doi:10.1214/aos/1018031270 Project Euclid: euclid.aos/1018031270 · Zbl 0951.62014 [37] Pericchi, L. R., Sansó, B. and Smith, A. F. M. (1993). Posterior cumulant relationships in Bayesian inference involving the exponential family. J. Amer. Statist. Assoc. 88 1419-1426. Zentralblatt MATH: 0794.62018 Digital Object Identifier: doi:10.1080/01621459.1993.10476427 · Zbl 0794.62018 [38] Proschan, F. (1963). Theoretical explanation of bbserved decreasing failure rate. Technometrics 5 375-383. [39] Reid, N. (2003). Asymptotics and the theory of inference. Ann. Statist. 31 1695-1731. Zentralblatt MATH: 1042.62022 Digital Object Identifier: doi:10.1214/aos/1074290325 Project Euclid: euclid.aos/1074290325 · Zbl 1042.62022 [40] Ruli, E., Sartori, N. and Ventura, L. (2014). Marginal posterior simulation via higher-order tail area approximations. Bayesian Anal. 9 129-145. Zentralblatt MATH: 1327.62159 Digital Object Identifier: doi:10.1214/13-BA851 · Zbl 1327.62159 [41] Speed, T. P. (1983). Cumulants and partition lattices. Aust. J. Stat. 25 378-388. Zentralblatt MATH: 0538.60023 Digital Object Identifier: doi:10.1111/j.1467-842X.1983.tb00391.x · Zbl 0538.60023 [42] Stuart, A. and Ord, J. K. (1994). Kendall’s Advanced Theory of Statistics, Vol. 1, 6th ed. Wiley, New York. Zentralblatt MATH: 0880.62012 · Zbl 0880.62012 [43] Sweeting, T. J. (1995). A framework for Bayesian and likelihood approximations in statistics. Biometrika 82 1-23. Zentralblatt MATH: 0829.62003 Digital Object Identifier: doi:10.1093/biomet/82.1.1 · Zbl 0829.62003 [44] Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. J. Amer. Statist. Assoc. 81 82-86. Zentralblatt MATH: 0587.62067 Digital Object Identifier: doi:10.1080/01621459.1986.10478240 · Zbl 0587.62067 [45] Tierney, L., Kass, R. E. and Kadane, J. B. (1989). Fully exponential Laplace approximations to expectations and variances of nonpositive functions. J. Amer. Statist. Assoc. 84 710-716. Zentralblatt MATH: 0682.62012 Digital Object Identifier: doi:10.1080/01621459.1989.10478824 · Zbl 0682.62012 [46] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge. [47] Ventura, L. and Reid, N. (2014). Approximate Bayesian computation with modified log-likelihood ratios. Metron 72 231-245. Zentralblatt MATH: 1316.62043 Digital Object Identifier: doi:10.1007/s40300-014-0041-4 · Zbl 1316.62043 [48] Wallace, D. L. (1958). Asymptotic approximations to distributions. Ann. Math. Stat. 29 635-654. Zentralblatt MATH: 0086.34004 Digital Object Identifier: doi:10.1214/aoms/1177706528 Project Euclid: euclid.aoms/1177706528 · Zbl 0086.34004 [49] Weng, R. C. (2003). On Stein’s identity for posterior normality. Statist. Sinica 13 495-506. Zentralblatt MATH: 1015.62087 · Zbl 1015.62087 [50] Weng, R. C. (2010). A Bayesian Edgeworth expansion by Stein’s identity. Bayesian Anal. 5 741-763. Zentralblatt MATH: 1330.62084 Digital Object Identifier: doi:10.1214/10-BA526 · Zbl 1330.62084 [51] Weng, R. C. and Tsai, W.-C. (2008). Asymptotic posterior normality for multiparameter problems. J. Statist. Plann. Inference 138 4068-4080. Zentralblatt MATH: 1146.62021 Digital Object Identifier: doi:10.1016/j.jspi.2008.03.034 · Zbl 1146.62021 [52] Withers, C. S. (1982). Second order inference for asymptotically normal random variables. Sankhyā, Ser. B 44 19-27. Zentralblatt MATH: 0549.62027 · Zbl 0549.62027 [53] Withers, C. S. (1984). Asymptotic expansions for distributions and quantiles with power series cumulants. J. Roy. Statist. Soc. Ser. B 46 389-396. Zentralblatt MATH: 0586.62026 Digital Object Identifier: doi:10.1111/j.2517-6161.1984.tb01310.x · Zbl 0586.62026 [54] Woodroofe, M. (1989). Very weak expansions for sequentially designed experiments: Linear models. Ann. Statist. 17 1087-1102. Zentralblatt MATH: 0683.62039 Digital Object Identifier: doi:10.1214/aos/1176347257 Project Euclid: euclid.aos/1176347257 · Zbl 0683.62039 [55] Woodroofe, M. · Zbl 0820.62019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.