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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.

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