Vine copulas as a mean for the construction of high dimensional probability distribution associated to a Markov Network

Building higher-dimensional copulas is generally recognized as a difficult problem. Regular-vines using bivariate copulas provide a flexible class of high-dimensional dependency models. In large dimensions, the drawback of the model is the exponentially increasing complexity. Recognizing some of the conditional independences is a possibility for reducing the number of levels of the pair-copula decomposition, and hence to simplify its construction (see [1]). The idea of using conditional independences was already performed under elliptical copula assumptions [17], [24] and in the case of DAGs in a recent work [2]. We provide a method which uses some of the conditional independences encoded by the Markov network underlying the variables. We give a theorem which under some graph conditions makes possible to derive pair-copula decomposition of the probability density function associated to a Markov network. As the underlying Markov network is usually unknown, we first have to discover it from the sample data. Using our results published in [33] and [21] we will show how to derive a multidimensional copula model exploiting the information on conditional independences hidden in the sample data.