Probability density decomposition for conditionally dependent random variables modeled by Vines

T.J. Bedford, R. Cooke

Research output: Contribution to journalArticle

331 Citations (Scopus)

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 lsquocanonical vinesrsquo built on highest degree trees offer the most efficient structure for Gibbs sampling.
Language English 245-268 23 Annals of Mathematics and Artificial Intelligence 32 1 10.1023/A:1016725902970 Published - 2001

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Dependent Random Variables
Probability Density
Random variables
Gibbs Sampling
Sampling
Decomposition
Decompose
Generalise
Conditional Independence
Dependent
Decomposition Theorem
Multivariate Distribution
Graphical Models
Clique
Modeling
Beliefs

Keywords

• probability
• statistics
• vine dependence
• markov tree
• management theory

Cite this

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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 lsquocanonical vinesrsquo built on highest degree trees offer the most efficient structure for Gibbs sampling.",
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In: Annals of Mathematics and Artificial Intelligence, Vol. 32, No. 1, 2001, p. 245-268.

Research output: Contribution to journalArticle

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AU - Cooke, R.

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KW - statistics

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KW - markov tree

KW - management theory

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