## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 11, Number 4 (2017), 2222-2251.

### Bayesian inference for multiple Gaussian graphical models with application to metabolic association networks

Linda S. L. Tan, Ajay Jasra, Maria De Iorio, and Timothy M. D. Ebbels

#### Abstract

We investigate the effect of cadmium (a toxic environmental pollutant) on the correlation structure of a number of urinary metabolites using Gaussian graphical models (GGMs). The inferred metabolic associations can provide important information on the physiological state of a metabolic system and insights on complex metabolic relationships. Using the fitted GGMs, we construct differential networks, which highlight significant changes in metabolite interactions under different experimental conditions. The analysis of such metabolic association networks can reveal differences in the underlying biological reactions caused by cadmium exposure. We consider Bayesian inference and propose using the multiplicative (or Chung–Lu random graph) model as a prior on the graphical space. In the multiplicative model, each edge is chosen independently with probability equal to the product of the connectivities of the end nodes. This class of prior is parsimonious yet highly flexible; it can be used to encourage sparsity or graphs with a pre-specified degree distribution when such prior knowledge is available. We extend the multiplicative model to multiple GGMs linking the probability of edge inclusion through logistic regression and demonstrate how this leads to joint inference for multiple GGMs. A sequential Monte Carlo (SMC) algorithm is developed for estimating the posterior distribution of the graphs.

#### Article information

**Source**

Ann. Appl. Stat. Volume 11, Number 4 (2017), 2222-2251.

**Dates**

Received: April 2016

Revised: June 2017

First available in Project Euclid: 28 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1514430284

**Digital Object Identifier**

doi:10.1214/17-AOAS1076

**Keywords**

Gaussian graphical models prior specification multiplicative model sequential Monte Carlo

#### Citation

Tan, Linda S. L.; Jasra, Ajay; De Iorio, Maria; Ebbels, Timothy M. D. Bayesian inference for multiple Gaussian graphical models with application to metabolic association networks. Ann. Appl. Stat. 11 (2017), no. 4, 2222--2251. doi:10.1214/17-AOAS1076. https://projecteuclid.org/euclid.aoas/1514430284

#### Supplemental materials

- Supplement to “Bayesian inference for multiple Gaussian graphical models with application to metabolic association networks”. We provide additional material to support the results in this paper. This include Matlab code, further discussions, detailed derivations and further results on the application to urinary metabolic data.Digital Object Identifier: doi:10.1214/17-AOAS1076SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.