# Phase marginal for a multivariate complex Gaussian density

Suppose $$z$$ is a random variable taking values in $$\mathbb{C}^n$$ and admitting the complex Gaussian density $$p(z;W) \propto \exp{(-\frac{1}{2}z^*Wz)}$$, where $$W$$ is Hermitian. Let $$r$$ be the vector of moduli and $$\theta$$ be the vector of phases; e.g., $$r_j = |z_j|$$ and $$\theta_j =$$ arg$$(z_j)$$.

I would like to compute or at least approximate the marginal density $$p(\theta; W)$$. Andersen et al.'s book on the complex normal distribution provides results for marginals on partitions of $$z$$, but nothing else, as far as I can tell. Other than that, I have only found a reference to the $$r$$ marginal, which apparently follows a multivariate central chi-square distribution with correlations induced by $$W$$ and has no closed form.

The integral $$\int_r p(r, \theta ;W) \ dr$$ seems difficult to calculate because of the couplings created by $$W$$ prevent me from factoring the density. I could diagonalize $$W$$, but I want to ensure this won't interfere with my ultimate goal, approximating the MLE for $$W$$ over some data set via gradient descent.

Any resources or suggestions for performing this calculation would be greatly appreciated. Thank you in advance.