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.
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References
T. Bedford and R.M. Cooke, Vines-a new graphical model for dependent random variables, Preprint, Delft University of Technology (1999), provisionally accepted for Annals of Statistics.
R.M. Cooke, A.M.H. Meeuwissen and C. Preyssl, Modularizing fault tree uncertainty analysis: The treatment of dependent information sources, in: Probabilistic Safety Assessment and Management, ed. G. Apostolakis (Elsevier, 1991).
R.M. Cooke, UNICORN: Methods and Code for Uncertainty Analysis (AEA Technologies, Warrington, 1995).
R.M. Cooke, Markov and entropy properties of tree-and vine-dependent variables, in: Proceedings of the ASA Section on Bayesian Statistical Science (1997).
J.L. Doob, Stochastic Processes (Wiley, 1953).
D. Gamerman, Markov Chain Monte Carlo; Stochastic Simulation for Bayesian Inference (Chapman and Hall, London, 1997).
S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Trans. Pattern. Anal. Mach. Intell. 6 (1984) 721-741.
R. Iman, J. Helton and J. Campbell, An approach to sensitivity analysis of computer models: Parts I and II, J. Quality Technol. 13(4) (1981).
R. Iman and W. Conover, A distribution-free approach to inducing rank correlation among input variables, Comm. Statist. Simulation Comput. 11(3) (1982) 311-334.
F.J. Jensen, An Introduction to Bayesian Networks (UCL Press, London, 1996).
H. Joe, Families of m-variate distributions with given margins and m(m ? 1)/2 bivariate dependence parameters, in: Distributions with Fixed Marginals and Related Topics, IMS Lecture Notes Monograph Series, Vol. 28, eds. L. Rüschendorf, B. Schweizer and M.D. Taylor (1996) pp. 120-141.
S. Kullback, Information Theory and Statistics (Wiley, 1959).
D. Kurowicka, Techniques in representing high dimensional distributions, Ph.D. Thesis, Delft University of Technology (2001).
A.M.H. Meeuwissen, Dependent random variables in uncertainty analysis, Ph.D. Thesis, Delft University of Technology (1993).
A.M.H. Meeuwissen and R.M. Cooke, Tree dependent random variables, Report 94-28, Department of Mathematics, Delft University of Technology (1994). 268 T. Bedford, R.M. Cooke / Conditionally dependent random variables modeled by vines
A.M.H. Meeuwissen and T.J. Bedford, Minimal informative distributions with given rank correlation for use in uncertainty analysis, J. Statist. Comput. Simulation 57(1-4) (1997) 143-175.
R.B. Nelsen, An Introduction to Copulas, Springer Lecture Notes in Statistics, Vol. 139 (Springer, New York, 1999).
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Bedford, T., Cooke, R.M. Probability Density Decomposition for Conditionally Dependent Random Variables Modeled by Vines. Annals of Mathematics and Artificial Intelligence 32, 245–268 (2001). https://doi.org/10.1023/A:1016725902970
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DOI: https://doi.org/10.1023/A:1016725902970