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

TL;DR

This paper investigates the distributional transform approximation for Gaussian copula models with discrete margins, demonstrating its near-exactness in certain variants and providing diagnostics to assess its suitability for inference.

Contribution

It proves the near-exactness of the distributional transform approach in specific models and introduces a diagnostic tool for practitioners to evaluate its applicability.

Findings

Distributional transform approximation is nearly exact in some model variants.

A diagnostic measure can assess the approximation's accuracy on observed data.

Practitioners can determine when genuine inference is feasible using this method.

Abstract

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.