Boundary-mode approximations for posterior expectations
Laplacian approximations to posterior expectations of smooth functions of parameters are considered when the posterior mode lies on the boundary of the parameter space. Since posterior expectations can be written as ratios of integrals, a simple approximation to the expectation can be obtained by approximating the numerator and the denominator integrals separately and then taking the ratio. Depending on the behavior of the gradient vectors of the numerator and the denominator integrands on the boundary, several types of approximations are developed having errors of different orders. The accuracies of these approximations are illustrated by one-and two-dimensional examples.