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

Abstract

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.

Bayesian learning of multiple directed networks from observational data

“…This results in a collection of distinct data sets. Some of these can be related to interventions on observed variables and were analyzed by Castelletti and Consonni 43 to infer a unique graph called interventional essential graph which reflects modifications in the edge structure due to interventions on nodes. The same data set was instead analyzed by Peterson et al 9 from a multiple undirected graphs perspective.…”

Discovering Causal Structures in Bayesian Gaussian Directed Acyclic Graph Models

“…The sequential computation of posterior edge inclusion probabilities requires the posterior probability distribution on the space of Markov equivalence classes conditionally on the data (observational and interventional) that are available at each step. To approximate the posterior probability distribution, this paper makes use of the OBIES procedure that was presented in Castelletti and Consonni (2019), which is based on an objective Bayes model comparison approach coupled with a standard binomial-beta prior on model space (Scott and Berger, 2010). Both assumptions could be easily relaxed, however, if subjective information is available (Ness et al, 2017), because all that our method requires as input is a posterior distribution on the space of essential graphs.…”

“…Besides in the original work Sachs et al (2005), the task of inferring the network underlying the whole Sachs data set was performed, using a variety of methods, in a series of papers including Friedman et al (2008), Luo and Zhao (2011), Hauser and Bühlmann (2015), Peterson et al (2015) and Castelletti and Consonni (2019), and the method of Castelletti and Consonni (2019) using a Bayesian approach.…”

“…Section 4 introduces the notion of an EG as the representative of a Markov equivalence class and clarifies how an intervention modifies its structure. Section 5 summarizes the objective Bayes methodology for structural learning of Gaussian essential graphical models in the presence of observational and interventional data (Castelletti and Consonni, 2019). Section 6 presents our active learning procedure for sequentially identifying optimal intervention variables.…”

Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures

“…Consequently, the identification of the true data‐generating DAG model can be improved in the presence of interventional data. An objective Bayes approach that jointly models observational and interventional data is also presented in Castelletti & Consonni (2019).…”