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Their marginal pdf is then (A.1).

The analytical integration of (4) with respect to the magnitude or phase to get the marginal PDF is extremely difficult or even impossible.

See Appendix B. Given the marginal CDF, the corresponding marginal PDF is easy to obtain, given the well-known result on the derivative of a determinant [52, Equation ], (14).

In this case, R1 has a uniform distribution in I and its (marginal) PDF is given by: f R 1 ( 1 ) ( r 1 ) = 1 z r 1 ∈ D 1 0 otherwise.

In the standard belief propagation (BP) algorithm (also known as sum-product), proposed by Pearl [20], the belief at node t, which represents the estimate of the posterior marginal PDF, is proportional to the product of the local evidence at that node ψ t (x t ), and all the messages coming into node t: M t ( x t ) ∝ ψ t ( x t ) ∏ u ∈ G t m ut ( x t ) (1).

In the context of BSS problem, the joint and the marginal PDF are unknown but they can be estimated.

In (A.5), the marginal pdf of is given by (A.6).

When and from (6), it follows that the series representation of the marginal pdf of is (A.7).

By definition, the mutual information of two random variables and with joint probability density function (pdf) and marginal pdfs and is the relative entropy between the joint distribution and the product distribution, that is, (14).

Therefore, the joint PDF is equal to the product of the marginal PDFs and the copula density function, i.e., the term c(F s(s),F n(n);θ c ).

It is shown in (3) that a multivariate PDF is constructed by multiplying a copula density function and a set of marginal PDFs in which the copula density function can be selected independent of the margins.