On the Occasional Exactness of the Distributional Transform Approximation for Direct Gaussian Copula Models with Discrete Margins

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.