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Laplace approximations to posterior moments and marginal distributions on circles, spheres, and cylinders
Author(s) -
Bagchi Parthasarathy,
Kadane Joseph B.
Publication year - 1991
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315537
Subject(s) - mathematics , laplace transform , laplace's method , spheres , mathematical analysis , laplace's equation , line (geometry) , approximations of π , real line , distribution (mathematics) , von mises yield criterion , geometry , physics , partial differential equation , astronomy , finite element method , thermodynamics
We extend recent work on Laplace approximations (Tierney and Kadane 1986; Tierney, Kass, and Kadane 1989) from parameter spaces that are subspaces of R k to those that are on circles, spheres, and cylinders. While such distributions can be mapped onto the real line (for example, a distribution on the circle can be thought of as a function of an angle θ, 0 ⩽ 0 ⩾ 2π), that the end points coincide is not a feature of the real line, and requires special treatment. Laplace approximations on the real line make essential use of the normal integral in both the numerator and the denominator. Here that role is played by the von Mises integral on the circle, by the Bingham integrals on the spheres and hyperspheres, and by the normal‐von Mises and normal‐Bingham integrals on the cylinders and hypercylinders, respectively. We begin with a brief introduction to Laplace approximations and to previous Bayesian work on circles, spheres, and cylinders. We then develop the theory for parameter spaces that are hypercylinders, since all other shapes considered here are special cases. We compute some examples, which show reasonable accuracy even for small samples.