Expectation Propagation for the Estimation of Conditional Bivariate Copulas
José Miguel Hernánez Lobato, David López-Paz and Zoubin Ghahramani
In: NIPS 2011 Workshop on Copulas in Machine Learning, 16 Dec 2011, Granada, Spain.
We present a semi-parametric method for the estimation of the copula of two random variables X and Y when conditioning to an additional covariate Z. The conditional bivariate copula is described using a parametric model fully specified in terms of Kendall's tau. The dependence of the conditional copula on Z is captured by expressing tau as a function of Z. In particular, tau is obtained by filtering a non-linear latent function, which is evaluated on Z, through a sigmoid-like function. A Gaussian process prior is assumed for the latent function and approximate Bayesian inference is performed using expectation propagation. A series of experiments with simulated and real-world data illustrate the advantages of the proposed approach.