Hyper inverse Wishart distribution for non-decomposable graphs and its application to Bayesian inference for Gaussian graphical models

Scandinavian Journal of Statistics

Volumn 29, Issue 3, 2002, Pages 391-411

  Roverato, Alberto ^a  

^a UNIVERSITY OF MODENA AND REGGIO EMILIA   (Italy)

Author keywords

Cholesky decomposition; Conjugate distribution; Gaussian graphical model; Hyper inverse Wishart distribution; Importance sampling; Matrix completion; Non decomposable graph; Normalizing constant

Indexed keywords

EID: 0035995077     PISSN: 03036898     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9469.00297     Document Type: Article

References (33)

1. Asmussen, S. &Edwards, D. (1983). Collapsibility and response variables in contingency tables. Biometrika 70, 567-578.
2. Bjerg, A.M. &Nielsen, T.H. (1993). Modelselektion i kovariansselektionsmodeller, MSc Thesis, Department of Mathematics and Computer Science, Aalborg University, Denmark.
3. Consonni, G. &Veronese, P. (2001). Conditionally reducible natural exponential families and enriched conjugate priors. Scand. J. Statist. 28, 377-406.
4. Cowell, R.G., Dawid, A.P., Lauritzen, S.L. &Spiegelhalter, D.J. (1999). Probabilistic networks and expert systems. Springer-Verlag, New York.
5. Cox, D.R. &Wermuth, N. (2000). On the generation of the chordless four-cycle. Biometrika 87, 206-212.
6. Dawid, A.P. (1981). Some matrix-variate distribution theory: notational considerations and a Bayesian application. Biometrika 68, 265-274.
7. Dawid, A.P. &Lauritzen, S.L. (1993). Hyper Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21, 1272-1317.
8. Dempster, A.P. (1972). Covariance selection. Biometrics 28, 157-175.
9. Diaconis, P. &Ylvisaker, D. (1979). Conjugate priors for exponential families. Ann. Statist. 7, 269-281.
10. Geiger, D. &Heckerman, D. (2000). Parameters priors for directed acyclic graphical models and the characterization of several probability distributions. Ann. Statist. to appear.
11. Giudici, P. (1996). Learning in graphical Gaussian models. In Bayesian Statistics 5 (eds J. Berger, J.M. Bernardo, A.P. Dawid &A.F.M. Smith), 621-628. Oxford University Press, Oxford.
12. Giudici, P. &Green, P.J. (1999). Decomposable graphical Gaussian model determination. Biometrika 86, 785-801.
13. Grone, R., Johnson, C.R., Sà, E.M. &Wolkowice, H. (1984). Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58, 109-124.
14. Gutiérrez-Peña, E. &Smith, A.F.M. (1997). Exponential and Bayesian conjugate families: review and extensions. Test 6, 1-90.
15. Isserlis, L. (1918). On a formula for the product-moment correlation of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134-139.
16. Lauritzen, S.L. (1996). Graphical models. Oxford University Press, Oxford.
17. Leimer, H.G. (1993). Optimal decomposition by clique separators. Discrete Math. 113, 99-123.
18. Massam, H. &Neher, E. (1997). On transformation and determinants of Wishart variables on symmetric cones. J. Theoret. Probab. 10, 867-902.
19. Massam, H. &Neher, E. (1998). Estimation and testing for lattice conditional independence models on euclidean Jordan algebras. Ann. Statist. 26, 1051-1082.
20. Muirhead, R.J. (1982). Aspects of multivariate statistical theory. Wiley, New York.
21. Paulsen, V.I., Power, S.C. &Smith, R.R. (1989). Schur products and matrix completions. J. Funct. Anal. 85, 151-178.
22. Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika 86, 677-690.
23. Roverato, A. (2000). Cholesky decomposition of a hyper inverse Wishart matrix. Biometrika 87, 99-112.
24. Roverato, A. &Whittaker, J. (1998). The Isserlis matrix and its application to non-decomposable graphical Gaussian models. Biometrika 85, 711-725.
25. Speed, T.P. &Kiiveri, H. (1986). Gaussian Markov distributions over finite graphs. Ann. Statist. 14, 138-150.
26. Tanner, M. (1996). Tools for statistical inference; 2nd edn. Springer-Verlag, New York.
27. Tierney, L. &Kadane, J.B. (1986). Accurate approximations for posterior moments and marginal densities. J. Amer. Statist. Assoc. 81, 82-86.
28. Wermuth, N. (1976). Analogies between multiplicative models in contingency tables and covariance selection. Biometrics 32, 95-108.
29. Wermuth, N. (1980). Linear recursive equations, covariance selection and path analysis. J. Amer. Statist. Assoc. 75, 963-972.
30. Wermuth, N. (1989). Modelling effects in multivariate normal distributions. Methodika III, 74-93.
31. Wermuth, N. &Cox, D.R. (1998). On association models defined over independence graphs. Bernoulli 3, 477-495.
32. Wermuth, N. &Cox, D.R. (2000). A sweep operator for triangular matrices and its statistical applications. Research report, ZUMA 00-04.
33. Whittaker, J. (1990). Graphical models in applied multivariate statistics. Wiley, Chichester.