The theorems and other results in the following paper are correct, as far as I know, but there is a step missing in the proof of Theorem 1. Kass, R.E., Tierney, L. and Kadane, J.B. (1990) The validity of posterior expansions based on Laplace's method, Essays in Honor of George Bernard, eds. S. Geisser, J.S. Hodges, S.J. Press, and A. Zellner, Amsterdam: North Holland, 473-488. The validity of posterior expansions based on Laplace's method must consider (1) local regions vanishingly close to the mode (where expansions take place), (2) distant regions (e.g., the exterior of a ball about the mode having fixed radius, which will have exponentially-decreasing probability), and (3) non-distant regions (e.g., the interior of the ball in (2)). In the Kass, Tierney, and Kadane paper we neglected to spell out what happens in (3). The argument is not hard, and depends on the assumed positive definiteness of the Hessian matrix. For those who would like details, the oversight was corrected in Theorem 2.2.13 and Lemma 2.2.16 of my book, Kass and Vos, "The Geometry of Asymptotic Inference" (Wiley). The context there is curved exponential families but the method is the same. |