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Biometrika 1989 76(3):425-433; doi:10.1093/biomet/76.3.425
© 1989 by Biometrika Trust
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Approximate marginal densities of nonlinear functions

LUKE TIERNEY, ROBERT E. KASS and JOSEPH B. KADANE

School of Statistics, University of Minnesota Minneapolis, Minnesota 55455, U.S.A.
Department of Statistics, Carnegie Mellon University Pittsburgh, Pennsylvania 15213, U.S.A.

This paper presents an asymptotic approximation for the marginal density of a nonlinear function g({theta}) that is applicable when the joint density of {theta} is dominated by a single mode and the Jacobian of g is of full rank near that mode. The approximation is based on Laplace's method and its asymptotic properties are similar to those of the saddlepoint approximation. The approximation is applied to the computation of a marginal posterior density, a marginal sampling density and a marginal density based on a multivariate saddlepoint approximation to a joint density.

Key Words: Asymptotic normality • Laplace's method • Saddlepoint method


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