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Biometrika 1993 80(4):731-740; doi:10.1093/biomet/80.4.731
© 1993 by Biometrika Trust
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On Bartlett adjustments for approximate Bayesian inference

THOMAS J. DICICCIO and STEVEN E. STERN

Department of Statistics, Stanford University Stanford, California 94305, U.S.A.

In wide generality, the posterior distributions of the likehood ratio statistic and the posterior ratio statistic are chi-squared to error of order O(n–1), where n is sample size. The error in the chi-squared approximation can be reduced to order O(n–2) by Bartlett correction. In this paper, explicit formulae are derived for the Bartlett adjustment factors of both statistics, and the derivations are based on the Tierney. Kass & Kadane (1989) asymptotic approximation for marginal posterior probability density functions. The use of numerical differentiation to facilitate calculation of the Bartlett adjustments is also described. Some applications are considered that concern inference about regression models from both complete and right-censored data.

Key Words: Asymptotic expansion • Bartlett adjustment factor • Chi-squared approximation • Exponential regression model • Highest posterior density region • Likehood ratio • Maximum likehood statistics • Normal regression model • Posterior ratio • Score statistic


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