Vines inference. (English) Zbl 1272.62040

Summary: In this paper, vines [R. M. Cooke, “Markov and entropy properties of tree- and vine-dependent variables”, in: Proceedings of the ASA section of Bayesian statistical science (1997); T. Bedford and R. M. Cooke, Ann. Stat. 30, No. 4, 1031–1068 (2002; Zbl 1101.62339)] are reviewed. We prove that the product of 1 minus the square of partial correlations on a vine equals the determinant of the correlation matrix. This is used in learning vines. In model learning we are interested in models incorporating maximal (conditional) independence with minimal disturbance. This leads us to search for regular vines whose associated factorization of the determinant is dominant in the sense of majorization. We compare this with the method of independence graphs [J. Whittaker, Graphical models in applied multivariate statistics. Chichester: John Wiley & Sons Ltd. xiv, 448p. (1990; Zbl 0732.62056)].


62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H99 Multivariate analysis
62A09 Graphical methods in statistics
05C05 Trees
05C90 Applications of graph theory