Vines--a new graphical model for dependent random variables

TLDR

Vines generalize Markov trees by relaxing conditional independence, enabling flexible modeling of high‑dimensional distributions useful for uncertainty analysis and sensitivity studies where expert‑elicited rank correlations inform the dependence structure. A new graphical model, called a vine, for dependent random variables is introduced. Vines specify multivariate distributions by defining marginals and their coupling, and a special case generalizes Joe’s construction to build a multivariate normal via unrestricted partial correlations between –1 and 1. Given expert‑elicited rank correlations, a minimum‑information vine distribution can be constructed easily, and sampling from it is nearly as fast as independent sampling.

Abstract

A new graphical model, called a vine, for dependent random variables is introduced. Vines generalize the Markov trees often used in modelling high-dimensional 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. Vines can be used to specify multivariate distributions in a straightforward way by specifying various marginal distributions and the ways in which these marginals are to be coupled. Such distributions have applications in uncertainty analysis where the objective is to determine the sensitivity of a model output with respect to the uncertainty in unknown parameters. Expert information is frequently elicited to determine some quantitative characteristics of the distribution such as (rank) correlations. We show that it is simple to construct a minimum information vine distribution, given such expert information. Sampling from minimum information distributions with given marginals and (conditional) rank correlations specified on a vine can be performed almost as fast as independent sampling. A special case of the vine construction generalizes work of Joe and allows the construction of a multivariate normal distribution by specifying a set of partial correlations on which there are no restrictions except the obvious one that a correlation lies between $-1$ and 1.