A note on Gauss—Hermite quadrature

For Gauss—Hermite quadrature, we consider a systematic method for transforming the variable of integration so that the integrand is sampled in an appropriate region. The effectiveness of the quadrature then depends on the ratio of the integrand to some Gaussian density being a smooth function, well approximated by a low-order polynomial. It is pointed out that, in this approach, order one Gauss-Hermite quadrature becomes the Laplace approximationmxHermite quadrature becomes the Laplace approximationmxHermite quadrature becomes the Laplace approximationmxHermite quadrature becomes the Laplace approximation. Thus the quadrature as implemented here can be thought of as a higher-order Laplace approximation.