Log normalizer for Multivariate Gaussian (Exponential Family Representation)

Question

I am searching for the log normalizer based on the natural parameters for the multivariate gaussian in the exponential family representation. For the univariate gaussian, it is given by $$ a(\eta) = \frac{-\eta_1^2}{4 \eta_2} - \frac{1}{2} log(-2\eta_2) $$ (See e.g. https://www.cs.princeton.edu/courses/archive/fall11/cos597C/lectures/exponential-families.pdf)

For the multivariate gaussian, however, I am completly stuck and couldn't find any ressources stating it. Can someone help me out?

Thanks in advance!

Answer 1

The log normalizer for the multivariate Gaussian is given by \begin{equation} F(\Theta) = \frac{1}{4} tr(\Theta_2^{-1} \Theta_1 \Theta_1^T) - \frac{1}{2} log | \Theta_2 | + \frac{\mathcal{D}}{2} log \pi \end{equation} with $\Theta = (\Theta_1, \Theta_2)$ being the natural parameters and $tr(\cdot)$ denoting the trace of a matrix.