  Copula models have become one of the most widely used tools in the applied
modelling of multivariate data. Similarly, Bayesian methods are increasingly
used to obtain efficient likelihood-based inference. However, to date, there
has been only limited use of Bayesian approaches in the formulation and
estimation of copula models. This article aims to address this shortcoming in
two ways. First, to introduce copula models and aspects of copula theory that
are especially relevant for a Bayesian analysis. Second, to outline Bayesian
approaches to formulating and estimating copula models, and their advantages
over alternative methods. Copulas covered include Archimedean, copulas
constructed by inversion, and vine copulas; along with their interpretation as
transformations. A number of parameterisations of a correlation matrix of a
Gaussian copula are considered, along with hierarchical priors that allow for
Bayesian selection and model averaging for each parameterisation. Markov chain
Monte Carlo sampling schemes for fitting Gaussian and D-vine copulas, with and
without selection, are given in detail. The relationship between the prior for
the parameters of a D-vine, and the prior for a correlation matrix of a
Gaussian copula, is discussed. Last, it is shown how to compute Bayesian
inference when the data are discrete-valued using data augmentation. This
approach generalises popular Bayesian methods for the estimation of models for
multivariate binary and other ordinal data to more general copula models.
Bayesian data augmentation has substantial advantages over other methods of
estimation for this class of models.
