Objective Bayes model selection of Gaussian interventional essential graphs for the identification of signaling pathways

A signalling pathway is a sequence of chemical reactions initiated by a stimulus which in turn affects a receptor, and then through some intermediate steps cascades down to the final cell response. Based on the technique of flow cytometry, samples of cell-by-cell measurements are collected under each experimental condition, resulting in a collection of interventional data (assuming no latent variables are involved). Usually several external interventions are applied at different points of the pathway, the ultimate aim being the structural recovery of the underlying signalling network which we model as a causal Directed Acyclic Graph (DAG) using intervention calculus. The advantage of using interventional data, rather than purely observational one, is that identifiability of the true data generating DAG is enhanced. More technically a Markov equivalence class of DAGs, whose members are statistically indistinguishable based on observational data alone, can be further decomposed, using additional interventional data, into smaller distinct Interventional Markov equivalence classes. We present a Bayesian methodology for structural learning of Interventional Markov equivalence classes based on observational and interventional samples of multivariate Gaussian observations. Our approach is objective, meaning that it is based on default parameter priors requiring no personal elicitation; some flexibility is however allowed through a tuning parameter which regulates sparsity in the prior on model space. Based on an analytical expression for the marginal likelihood of a given Interventional Essential Graph, and a suitable MCMC scheme, our analysis produces an approximate posterior distribution on the space of Interventional Markov equivalence classes, which can be used to provide uncertainty quantification for features of substantive scientific interest, such as the posterior probability of inclusion of selected edges, or paths.