The Annals of Statistics

Exact formulas for the normalizing constants of Wishart distributions for graphical models

Caroline Uhler, Alex Lenkoski, and Donald Richards

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Gaussian graphical models have received considerable attention during the past four decades from the statistical and machine learning communities. In Bayesian treatments of this model, the $G$-Wishart distribution serves as the conjugate prior for inverse covariance matrices satisfying graphical constraints. While it is straightforward to posit the unnormalized densities, the normalizing constants of these distributions have been known only for graphs that are chordal, or decomposable. Up until now, it was unknown whether the normalizing constant for a general graph could be represented explicitly, and a considerable body of computational literature emerged that attempted to avoid this apparent intractability. We close this question by providing an explicit representation of the $G$-Wishart normalizing constant for general graphs.

Article information

Ann. Statist., Volume 46, Number 1 (2018), 90-118.

Received: June 2014
Revised: January 2017
First available in Project Euclid: 22 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H05: Characterization and structure theory 60E05: Distributions: general theory
Secondary: 62E15: Exact distribution theory

Bartlett decomposition bipartite graph Cholesky decomposition chordal graph directed acyclic graph $G$-Wishart distribution Gaussian graphical model generalized hypergeometric function of matrix argument moral graph normalizing constant Wishart distribution


Uhler, Caroline; Lenkoski, Alex; Richards, Donald. Exact formulas for the normalizing constants of Wishart distributions for graphical models. Ann. Statist. 46 (2018), no. 1, 90--118. doi:10.1214/17-AOS1543.

Export citation


  • [1] Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and Its Applications 71. Cambridge Univ. Press, Cambridge.
  • [2] Atay-Kayis, A. and Massam, H. (2005). A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models. Biometrika 92 317–335.
  • [3] Bojdecki, T. and Gorostiza, L. G. (1999). Fractional Brownian motion via fractional Laplacian. Statist. Probab. Lett. 44 107–108.
  • [4] Cheng, Y. and Lenkoski, A. (2012). Hierarchical Gaussian graphical models: Beyond reversible jump. Electron. J. Stat. 6 2309–2331.
  • [5] Dawid, A. P. and Lauritzen, S. L. (1993). Hyper-Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21 1272–1317.
  • [6] Dempster, A. P. (1972). Covariance selection. Biometrics 28 157–175.
  • [7] Dobra, A. and Lenkoski, A. (2011). Copula Gaussian graphical models and their application to modeling functional disability data. Ann. Appl. Stat. 5 969–993.
  • [8] Dobra, A., Lenkoski, A. and Rodriguez, A. (2011). Bayesian inference for general Gaussian graphical models with application to multivariate lattice data. J. Amer. Statist. Assoc. 106 1418–1433.
  • [9] Gårding, L. (1947). The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 785–826.
  • [10] Giri, N. C. (2004). Multivariate Statistical Analysis. Dekker, New York.
  • [11] Giudici, P. and Green, P. J. (1999). Decomposable graphical Gaussian model determination. Biometrika 86 785–801.
  • [12] Gross, K. I. and Richards, D. St. P. (1987). Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 781–811.
  • [13] Haff, L. R., Kim, P. T., Koo, J.-Y. and Richards, D. St. P. (2011). Minimax estimation for mixtures of Wishart distributions. Ann. Statist. 39 3417–3440.
  • [14] Herz, C. S. (1955). Bessel functions of matrix argument. Ann. of Math. (2) 61 474–523.
  • [15] Hille, E. and Phillips, R. S. (1957). Functional Analysis and Semi-Groups, rev. ed. American Mathematical Society Colloquium Publications 31. Amer. Math. Soc., Providence, RI.
  • [16] Ingham, A. E. (1933). An integral which occurs in statistics. Math. Proc. Cambridge Philos. Soc. 29 271–276.
  • [17] James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Stat. 35 475–501.
  • [18] Jones, B., Carvalho, C., Dobra, A., Hans, C., Carter, C. and West, M. (2005). Experiments in stochastic computation for high-dimensional graphical models. Statist. Sci. 20 388–400.
  • [19] Karlin, S. (1968). Total Positivity, Vol. I. Stanford Univ. Press, Stanford, CA.
  • [20] Kim, P. T. and Richards, D. St. P. (2011). Deconvolution density estimation on the space of positive definite symmetric matrices. In Nonparametric Statistics and Mixture Models 147–168. World Sci. Publ., Hackensack, NJ.
  • [21] Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Clarendon Press, Oxford.
  • [22] Lenkoski, A. (2013). A direct sampler for $G$-Wishart variates. Stat 2 119–128.
  • [23] Lenkoski, A. and Dobra, A. (2011). Computational aspects related to inference in Gaussian graphical models with the G-Wishart prior. J. Comput. Graph. Statist. 20 140–157. Supplementary material available online.
  • [24] Letac, G. and Massam, H. (2007). Wishart distributions for decomposable graphs. Ann. Statist. 35 1278–1323.
  • [25] Maass, H. (1971). Siegel’s Modular Forms and Dirichlet Series. Lecture Notes in Mathematics 216. Springer, Berlin. Dedicated to the last great representative of a passing epoch Carl Ludwig Siegel on the occasion of his seventy-fifth birthday.
  • [26] Mitsakakis, N., Massam, H. and Escobar, M. D. (2011). A Metropolis–Hastings based method for sampling from the $G$-Wishart distribution in Gaussian graphical models. Electron. J. Stat. 5 18–30.
  • [27] Muirhead, R. J. (1975). Expressions for some hypergeometric functions of matrix argument with applications. J. Multivariate Anal. 5 283–293.
  • [28] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York.
  • [29] Olkin, I. (2002). The 70th anniversary of the distribution of random matrices: A survey. Linear Algebra Appl. 354 231–243. Ninth special issue on linear algebra and statistics.
  • [30] Piccioni, M. (2000). Independence structure of natural conjugate densities to exponential families and the Gibbs’ sampler. Scand. J. Stat. 27 111–127.
  • [31] Roverato, A. (2002). Hyper inverse Wishart distribution for non-decomposable graphs and its application to Bayesian inference for Gaussian graphical models. Scand. J. Stat. 29 391–411.
  • [32] Siegel, C. L. (1935). Über die analytische Theorie der quadratischen Formen. Ann. of Math. (2) 36 527–606.
  • [33] Speed, T. P. and Kiiveri, H. T. (1986). Gaussian Markov distributions over finite graphs. Ann. Statist. 14 138–150.
  • [34] Uhler, C., Lenkoski, A. and Richards, D. (2017). Supplement to “Exact formulas for the normalizing constants of Wishart distributions for graphical models.” DOI:10.1214/17-AOS1543SUPP.
  • [35] Wang, H. and Carvalho, C. M. (2010). Simulation of hyper-inverse Wishart distributions for non-decomposable graphs. Electron. J. Stat. 4 1470–1475.
  • [36] Wang, H. and Li, S. Z. (2012). Efficient Gaussian graphical model determination under $G$-Wishart prior distributions. Electron. J. Stat. 6 168–198.
  • [37] Wishart, J. (1928). The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A 32–52.
  • [38] Wishart, J. and Bartlett, M. S. (1933). The generalised product moment distribution in a normal system. Math. Proc. Cambridge Philos. Soc. 29 260–270.

Supplemental materials

  • Supplement to “Exact formulas for the normalizing constants of Wishart distributions for graphical models”. Exact formulas for the normalizing constants of Wishart distributions for graphical models with minimum fill-in 1.