# Principled estimate of joint PDF given marginals and first and second-order statistics

(Trying to solve the same underlying problem as this.)

Let's say we know the marginal probability density functions $$p_i(x)$$ of a set of zero-mean random variables $$\{X_i\}_{i=1}^N$$ as well as the second-order statistics $$C_{ij} = \mathbb E [X_i X_j^*]$$. Assume that $$X_i$$ are complex-valued circularly-symmetric random variables (i.e., real and imaginary parts are independent).

Is there a principled way of producing an estimate of the joint distribution $$p(x_1, x_2, \ldots, x_n)$$?

In the case where the marginals are unknown, the maximum entropy distribution is simply the multivariate normal for both the real and imaginary parts. Is there a similar process that allows one to account for the additional information provided by the marginals?