  The direct Gaussian copula model with discrete marginal distributions is an
appealing data-analytic tool but poses difficult computational challenges due
to its intractable likelihood. A number of approximations/surrogates for the
likelihood have been proposed, including the continuous extension-based
approximation (CE) and the distributional transform-based approximation (DT).
The continuous extension approach is exact up to Monte Carlo error but does not
scale well computationally. The distributional transform approach permits
efficient computation but offers no theoretical guarantee that it is exact. In
practice, though, the distributional transform-based approximate likelihood is
so very nearly exact for some variants of the model as to permit genuine
maximum likelihood or Bayesian inference. We demonstrate the exactness of the
distributional transform-based objective function for two interesting variants
of the model, and propose a quantity that can be used to assess exactness for
experimentally observed datasets. Said diagnostic will permit practitioners to
determine whether genuine Bayesian inference or ordinary maximum likelihood
inference using the DT-based likelihood is possible for a given dataset.
