derivation

Consider Taylor series , where

Define foo31 and foo32 below, which will satisfy as shown later.

Define a rule to make , thus ih2 is the inverse matrix of h2.

Apply this rule to foo31+foo32 to get foo33.

Confirming foo33=

Since is the multivariate normal density function with mean specified above and covariance, =. Then, ψ(θ)=log

Considering , we find that and from the above expression of .  Let us write with

Rewrite the Taylor series of h(y) by noting the first derivative h1 is in fact .

In the following, , where .

Noting , where the expectation is taken for the multivariate normal with mean and covariance identity. We apply "logeexppoly" to foo37. First, we get foo38={a0,a1,a2,a3,a4} for the coefficients of , and below.

Now substitute for in foo39 to get ψ(θ)=foo40 in terms of η below. The coefficients for the polynomial of η is stored in foo41.

Since foo40=ψ(θ)=psieta, foo41 must be equal to foo42 below.

At first, we compare the coefficient for .

Thus, ah2 is in fact instead of .

We rewrite foo43=foo42-foo41.

We solve these five equations ⩵ 0. At first, foo43[[4]]==0 gives