  Copulas provide a modular parameterization of multivariate distributions that
decouples the modeling of marginals from the dependencies between them.
Gaussian Mixture Copula Model (GMCM) is a highly flexible copula that can model
many kinds of multi-modal dependencies, as well as asymmetric and tail
dependencies. They have been effectively used in clustering non-Gaussian data
and in Reproducibility Analysis, a meta-analysis method designed to verify the
reliability and consistency of multiple high-throughput experiments. Parameter
estimation for GMCM is challenging due to its intractable likelihood. The best
previous methods have maximized a proxy-likelihood through a Pseudo Expectation
Maximization (PEM) algorithm. They have no guarantees of convergence or
convergence to the correct parameters. In this paper, we use Automatic
Differentiation (AD) tools to develop a method, called AD-GMCM, that can
maximize the exact GMCM likelihood. In our simulation studies and experiments
with real data, AD-GMCM finds more accurate parameter estimates than PEM and
yields better performance in clustering and Reproducibility Analysis. We
discuss the advantages of an AD-based approach, to address problems related to
monotonic increase of likelihood and parameter identifiability in GMCM. We also
analyze, for GMCM, two well-known cases of degeneracy of maximum likelihood in
GMM that can lead to spurious clustering solutions. Our analysis shows that,
unlike GMM, GMCM is not affected in one of the cases.
