Dipartimento di Scienze Statistiche
Università Cattolica del Sacro Cuore
“Marginal Likelihood of Essential Gaussian Graphical Models: an Objective Bayes Approach”
Graphical models based on Directed Acyclic Graphs (DAGs) are widely used to model dependency relationships among a set of many variables in various scientific areas. We would like to learn the structure of the DAG underlying a sample of observations. However, it is well known that different DAGs can encode the same set of conditional independencies; as a consequence one cannot distinguish DAGs using observational data only. For this reason, we consider Markov equivalence classes of DAGs. Each class contains exactly those DAGs encoding the same conditional independencies. A Markov equivalence class can be represented by a single unique chain graph, called the essential graph (EG), which combines features of decomposable undirected graphs and DAGs. Our interest lies in graphical model selection on the space of EGs. To this end, we construct suitable objective priors, based on the fractional Bayes factor, and compute the marginal likelihood of an essential Gaussian graphical model.