## The Annals of Statistics

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

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

Caroline Uhler, Alex Lenkoski, and Donald Richards

#### Abstract

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

**Source**

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

**Dates**

Received: June 2014

Revised: January 2017

First available in Project Euclid: 22 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1519268425

**Digital Object Identifier**

doi:10.1214/17-AOS1543

**Mathematical Reviews number (MathSciNet)**

MR3766947

**Zentralblatt MATH identifier**

06865106

**Subjects**

Primary: 62H05: Characterization and structure theory 60E05: Distributions: general theory

Secondary: 62E15: Exact distribution theory

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1519268425

#### 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.Digital Object Identifier: doi:10.1214/17-AOS1543SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.