Approximating the marginal likelihood using copula

TL;DR

This paper introduces copula-based extensions of the Laplace approximation to efficiently estimate marginal likelihoods in Bayesian model selection, improving accuracy especially for complex dependencies.

Contribution

It presents novel copula-based methods for approximating marginal likelihoods, applicable with or without posterior simulation, and enhances accuracy using t-copulas for non-Gaussian dependencies.

Findings

Copula-based Laplace approximations are easy to implement.

The methods work with and without posterior simulation.

T-copulas improve accuracy for complex dependencies.

Abstract

Model selection is an important activity in modern data analysis and the conventional Bayesian approach to this problem involves calculation of marginal likelihoods for different models, together with diagnostics which examine specific aspects of model fit. Calculating the marginal likelihood is a difficult computational problem. Our article proposes some extensions of the Laplace approximation for this task that are related to copula models and which are easy to apply. Variations which can be used both with and without simulation from the posterior distribution are considered, as well as use of the approximations with bridge sampling and in random effects models with a large number of latent variables. The use of a t-copula to obtain higher accuracy when multivariate dependence is not well captured by a Gaussian copula is also discussed.