Probability Density Decomposition for Conditionally Dependent Random Variables Modeled by Vines

Authors:
Tim Bedford;Roger M. Cooke
Affiliations:
Department of Management Science, Strathclyde University, 40 George St, Glasgow G1 1QE, Scotland E-mail: tim@mansci.strath.ac.uk;Faculty of Information Technology and Systems, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands E-mail: r.m.cooke@its.tudelft.nl
Venue:
Annals of Mathematics and Artificial Intelligence
Year:
2001

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Abstract

A vine is a new graphical model for dependent random variables. Vines generalize the Markov trees often used in modeling multivariate distributions. They differ from Markov trees and Bayesian belief nets in that the concept of conditional independence is weakened to allow for various forms of conditional dependence. A general formula for the density of a vine dependent distribution is derived. This generalizes the well-known density formula for belief nets based on the decomposition of belief nets into cliques. Furthermore, the formula allows a simple proof of the Information Decomposition Theorem for a regular vine. The problem of (conditional) sampling is discussed, and Gibbs sampling is proposed to carry out sampling from conditional vine dependent distributions. The so-called ‘canonical vines’ built on highest degree trees offer the most efficient structure for Gibbs sampling.