Commentary - A Latent Variable Perspective of Copula Modeling

The Essential Ideas Let us begin by congratulating Danaher and Smith (2011) on an excellent contribution that serves as a lucid introduction to copula modeling, as well as providing a sensible Bayesian approach for its application to both continuous and discrete data. The essential concept we take away is that modeling dependence in multivariate data is facilitated by transforming the marginal data distributions to spaces where dependencies are more naturally represented. This central idea is most clearly illustrated in the case where each component of the original multivariate data !X1" # # # "Xp$ is a realization of a continuous random variable. Each continuous Xj with cumulative distribution function (cdf) Fj can be transformed to a desired random variable X j by first transforming Xj to a uniform random variable, Uj = Fj!Xj$" and then transforming this to X j =Gj !Uj$" where Gj is the inverse of the cdf F ⋆ j of X j . The goal is to select marginal distributions for X j that have a natural dependence structure. For example, if X j is chosen to have a Gaussian distribution, as Danaher and Smith recommend, then !X 1" # # # "X p$ are treated as a multivariate Gaussian vector with unknown covariance &" which could be estimated with the X j observations. The resulting dependence structure implicitly imposed on !U1" # # # "Up$ and hence on !X1" # # # "Xp$ is the Gaussian copula. As Danaher and Smith point out, this idea extends to the general case by treating discrete Xj as corresponding to a latent uniform variable Uj" which takes values according to their Equation (5). Equivalently, any random variable Xj , with cdf Fj , can be considered as the realization of an underlying uniform Uj via Xj =Gj!Uj$, where Gj!u$= inf'x( u≤ Fj!x$) (1)