Updating $q(s)$ for the Gaussian node

Here we show how to minimise the function

$${\cal C}(m,v) = Mm + V[(m-m_0)^2 + v] + E \exp(m+v/2) - \frac{1}{2} \ln v,$$ (54)

where $M$,$V$,$E$, and $m_0$ are scalar constants. A unique solution exists when $V > 0$ and $E \geq 0$. This problem occurs when a Gaussian posterior with mean $m$ and variance $v$ is fitted to a probability distribution whose logarithm has both a quadratic and exponential part resulting from Gaussian prior and log-Gamma likelihoods, respectively, and Kullback-Leibler divergence is used as the measure of the misfit.

In the special case $E = 0$, the minimum of ${\cal C}(m,v)$ can be found analytically and it is $m=m_0-\frac{M}{2V}$, $v=\frac{1}{2V}$. In other cases where $E>0$, minimisation is performed iteratively. At each iteration, one Newton iteration for the mean $m$ and one fixed-point iteration for the variance $v$ are carried out as explained in more detail in the following.