Learning Markov Equivalence Classes of Directed Acyclic Graphs: An Objective Bayes Approach

Castelletti, Federico; Consonni, Guido; Vedova, Marco L. Della; Peluso, Stefano

doi:10.1214/18-ba1101

Cited by 29 publications

(58 citation statements)

References 35 publications

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“…These can be obtained using standard conjugate priors based on normal and normal‐Wishart distributions, thus avoiding more complex priors specifically targeted to ( constrained ) DAG or EG models. A detailed discussion of these results can be found in Consonni & La Rocca (2012) and Castelletti et al (2018) where two objective Bayes approaches for model selection of DAG models and EGs are presented.…”

Section: Priors For Graphical Model Comparisonmentioning

confidence: 92%

“…Finally, to construct an MH MCMC scheme on the model space

{scriptS}_{q}

, we need to assign a prior on

scriptG \in {scriptS}_{q}

. In the following, we adopt a very simple prior that only depends on the number of edges in the graph (equivalently, on its skeleton) (Castelletti et al , 2018). Consequently, such prior can be used for both DAGs and EGs and assigns the same probability to each pair of graphs having the same number of edges.…”

Section: Markov Chain Monte Carlo Methodsmentioning

confidence: 99%

“…Codes and additional details can be found at https://github.com/FedeCastelletti/obayes_learn_essential_graphs. Codes refer to the MCMC presented in Castelletti et al (2018), which follows the structure of Algorithms 2 and 3 using the marginal likelihood of an EG (14) computed through the fractional Bayes factor approach (see also the original paper for details).…”

Section: Markov Chain Monte Carlo Methodsmentioning

confidence: 99%

“…Through this paper, we aim at reviewing some contributions to Bayesian model selection of Gaussian DAG and EG models in a unified and comprehensive framework; we refer in particular to the works Consonni & La Rocca (2012), Consonni et al (2017) and Castelletti et al (2018). All of these contributions are targeted to (objective Bayes) model selection of graphical models via marginal likelihood , which in a Bayesian context represents the score assigned to a given model.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, we present a general and flexible MCMC strategy that can be adopted for both model selection of DAG and EG. More in details, we first focus on prior specification for model parameters, relying on the methodology proposed by Geiger & Heckerman (2002) and then adopted by Consonni & La Rocca (2012) and Castelletti et al (2018) for objective model selection of Gaussian DAG and EG models, respectively. As discussed in Section 3, this approach relies on a set of assumptions that are naturally satisfied in the Gaussian setting and provide an effective solution to the above‐mentioned issue of prior elicitation.…”

Section: Introductionmentioning

confidence: 99%

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Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures

Castelletti

2020

Int Statistical Rev

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Summary During the last years, graphical models have become a popular tool to represent dependencies among variables in many scientific areas. Typically, the objective is to discover dependence relationships that can be represented through a directed acyclic graph (DAG). The set of all conditional independencies encoded by a DAG determines its Markov property. In general, DAGs encoding the same conditional independencies are not distinguishable from observational data and can be collected into equivalence classes, each one represented by a chain graph called essential graph (EG). However, both the DAG and EG space grow super exponentially in the number of variables, and so, graph structural learning requires the adoption of Markov chain Monte Carlo (MCMC) techniques. In this paper, we review some recent results on Bayesian model selection of Gaussian DAG models under a unified framework. These results are based on closed‐form expressions for the marginal likelihood of a DAG and EG structure, which is obtained from a few suitable assumptions on the prior for model parameters. We then introduce a general MCMC scheme that can be adopted both for model selection of DAGs and EGs together with a couple of applications on real data sets.

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Section: Priors For Graphical Model Comparisonmentioning

confidence: 92%

“…Finally, to construct an MH MCMC scheme on the model space

{scriptS}_{q}

, we need to assign a prior on

scriptG \in {scriptS}_{q}

Section: Markov Chain Monte Carlo Methodsmentioning

confidence: 99%

Section: Markov Chain Monte Carlo Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures

Castelletti

2020

Int Statistical Rev

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Bayesian learning of multiple directed networks from observational data

Castelletti

Rocca

Peluso

et al. 2020

Statistics in Medicine

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Graphical modeling represents an established methodology for identifying complex dependencies in biological networks, as exemplified in the study of co‐expression, gene regulatory, and protein interaction networks. The available observations often exhibit an intrinsic heterogeneity, which impacts on the network structure through the modification of specific pathways for distinct groups, such as disease subtypes. We propose to infer the resulting multiple graphs jointly in order to benefit from potential similarities across groups; on the other hand our modeling framework is able to accommodate group idiosyncrasies. We consider directed acyclic graphs (DAGs) as network structures, and develop a Bayesian method for structural learning of multiple DAGs. We explicitly account for Markov equivalence of DAGs, and propose a suitable prior on the collection of graph spaces that induces selective borrowing strength across groups. The resulting inference allows in particular to compute the posterior probability of edge inclusion, a useful summary for representing flow directions within the network. Finally, we detail a simulation study addressing the comparative performance of our method, and present an analysis of two protein networks together with a substantive interpretation of our findings.

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