Copula functions for learning multimodal densities with non-linear dependencies

In this work, we propose a new framework for learning mixture models from continuous data. Gaussian Mixture Models (GMMs) are commonly used for this task and are popular among practitioners because of their sound statistical foundation and the availability of an efficient learning algorithm [2]. However, the underlying assumption about the normally distributed mixing components, is often too rigid for several real life datasets. With the aim of relaxing this assumption, we introduce a new class of parametric mixture models whose foundation is laid on the theory of Copula functions. Copula functions provide an elegant way of modeling joint densities of random variables by providing explicit control on the form of univariate marginals [5, 3, 4]. As a result, the overall joint density can be written as a product of marginal densities with a density function (copula density) that encodes the dependencies between the random variables.