The Annals of Statistics

Vines--a new graphical model for dependent random variables

Tim Bedford and Roger M. Cooke

Full-text: Open access


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.

Article information

Ann. Statist., Volume 30, Number 4 (2002), 1031-1068.

First available in Project Euclid: 10 September 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E10: Characterization and structure theory 62E25 62H20: Measures of association (correlation, canonical correlation, etc.) 62B10: Information-theoretic topics [See also 94A17] 94A17: Measures of information, entropy

Correlation dependence information multivariate probablility distribution Monte Carlo simulation tree dependence Markov tree belief net multivariate normal distribution


Bedford, Tim; Cooke, Roger M. Vines--a new graphical model for dependent random variables. Ann. Statist. 30 (2002), no. 4, 1031--1068. doi:10.1214/aos/1031689016.

Export citation


  • [1] ARNOLD, B. C., CASTILLO, E. and SARABIA, J. M. (1999). Conditional Specification of Statistical Models. Springer, New York.
  • [2] COOKE, R. M. (1995). UNICORN: Methods and Code for Uncertainty Analy sis. Atomic Energy Association, Delft Univ. Technology.
  • [3] COOKE, R. M. (1997). Markov and entropy properties of treeand vine-dependent variables. Proc. ASA Section on Bayesian Statistical Science 166-175. Amer. Statist. Assoc., Alexandria, VA.
  • [4] COOKE, R. M., KEANE, M. S. and MEEUWISSEN, A. M. H. (1990). User's manual for RIAN: Computerized risk assessment. Estec 1233, Delft Univ. Technology.
  • [5] COOKE, R. M., MEEUWISSEN, A. M. H. and PREy SSL, C. (1991). Modularizing fault tree uncertainty analysis: The treatment of dependent information sources. In Probabilistic Safety Assessment and Management (G. Apostolakis, ed.). North-Holland, Amsterdam.
  • [6] CUADRAS, C. M. (1992). Probability distributions with given multivariate marginals and given dependence structure. J. Multivariate Anal. 42 51-66.
  • [7] GENEST, C., QUESADA, M., JOSÉ, J., RODRIGUEZ, L. and JOSÉ, A. (1995). De l'impossibilité de construire des lois à marges multidimensionnelles données à partir de copules. C. R. Acad. Sci. Paris Sér. I Math. 320 723-726.
  • [8] GUIA ¸SU, S. (1977). Information Theory with Applications. McGraw-Hill, New York.
  • [9] O'HAGAN, A. (1994). Kendall's Advanced Theory of Statistics 2B. Bayesian Inference. Arnold, London.
  • [10] IMAN, R. and CONOVER, W. (1982). A distribution-free approach to inducing rank correlation among input variables. Comm. Statist. Simulation Comput. 11 311-334.
  • [11] IMAN, R., HELTON, J. and CAMPBELL, J. (1981). An approach to sensitivity analysis of computer models. I, II. J. Quality Technology 13 174-183, 232-240.
  • [12] JAy NES, E. T. (1983). Papers on Probability, Statistics and Statistical physics (R. D. Rosenkrantz, ed.). Reidel, Dordrecht.
  • [13] JOE, H. (1996). Families of m-variate distributions with given margins and m(m 1)/2 bivariate dependence parameters. In Distributions with Fixed Marginals and Related Topics (L. Rüschendorf, B. Schweizer and M. D. Tay lor, eds.) 120-141. IMS, Hay ward, CA.
  • [14] JOE, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
  • [15] KENDALL, M. G. and STUART, A. (1967). The Advanced Theory of Statistics 2. Inference and Relationship, 2nd ed. Griffin, London.
  • [16] KRAAN, B. (2001). Probabilistic inversion in uncertainty analysis. Ph.D. thesis, Delft Univ. Technology.
  • [17] KULLBACK, S. (1959). Information Theory and Statistics. Wiley, New York.
  • [18] LI, H., SCARSINI, M. and SHAKED, M. (1996). Linkages: a tool for the construction of multivariate distributions with given non-overlapping multivariate marginals. J. Multivariate Anal. 56 20-41.
  • [19] LI, H., SCARSINI, M. and SHAKED, M. (1999). Dy namic linkages for multivariate distributions with given non-overlapping multivariate marginals. J. Multivariate Anal. 68 54-77.
  • [20] MEEUWISSEN, A. M. H. (1993). Dependent random variables in uncertainty analysis. Ph.D. thesis, Delft Univ. Technology.
  • [21] MEEUWISSEN, A. M. H. and BEDFORD, T. J. (1997). Minimally informative distributions with given rank correlation for use in uncertainty analysis. J. Statist. Comput. Simulation 57 143-175.
  • [22] MEEUWISSEN, A. M. H. and COOKE, R. M. (1994). Tree dependent random variables. Technical Report 94-28, Dept. Mathematics, Delft Univ. Technology.
  • [23] MUIRHEAD, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • [24] OLIVER, R. M. and SMITH, J. Q. (eds.) (1990). Influence Diagrams, Belief Nets and Decision Analy sis. Wiley, Chichester.