The Vine Copula Method for Representing High Dimensional Dependent Distributions: Application to Continuous Belief Nets

High dimensional probabilistic models are often formulated as belief nets (BN’s), that is, as directed acyclic graphs with nodes representing random variables and arcs representing “influence”. BN’s are conditionalized on incoming information to support probabilistic inference in expert system applications. For continuous random variables, an adequate theory of BN’s exists only for the joint normal distribution. In general, an arbitrary correlation matrix is not compatible with arbitrary marginals, and conditionalization is quite intractable. Transforming to normals is unable to reproduce exactly a specified rank correlation matrix. We show that a continuous belief net can be represented as a regular vine, where an arc from node i to j is associated with a (conditional) rank correlation between i and j. Using the elliptical copula and the partial correlation transformation properties, it is very easy to conditionalize the distribution on the value of any node, and hence update the BN.