In the previous post, I alluded to using Hermite polynomials in conjunction with higher-order Laplace approximation. In this post I’ll expand on what that means.
Hermite polynomials are orthogonal polynomials over the real line with respect to the weight given by the standard normal distribution. (There are two conventions for defining Hermite polynomials, what Wikipedia calls the physicist convention and the probabilist convention. We’re using the latter here.)
The first few Hermite polynomials are 1, x, x2 – 1, x3 – 3x, … . You can find the rest of the Hermite polynomials via the recurrence relation
Hn+1(x) = x Hn – n Hn-1(x).
The odd order Hermite polynomials are odd functions, and the standard normal density is an even function, so the integral of their product is zero. For an even number m, the integral of the mth order Hermite polynomial times the standard normal density is (m-1)!!, i.e. m double factorial. In probability terms, if X is a standard normal random variable, the expected value of Hk(X) is 0 if k is odd and (k – 1)!! otherwise.
You could think of the Hermite polynomials as the right basis to use when working with normal probability distributions. Writing a polynomial as a linear combination of Hermite polyn0mials is a change of basis that makes integration easy:
Here [k even] is the function that returns 1 if k is even and 0 otherwise. This notation was introduced by Kenneth Iverson in the APL programming language and has become moderately common in mathematics.