Bayesian Networks are probabilistic graphical models that represent conditional independence relationships among variables as a directed acyclic graph (DAG), where edges can be interpreted as causal effects connecting one causal symptom to an effect symptom. These models can help overcome one of the key limitations of partial correlation networks whose edges are undirected. This tutorial aims to introduce Bayesian Networks to identify admissible causal relationships in cross-sectional data, as well as how to estimate these models in R through three algorithm families with an empirical example data set of depressive symptoms. In addition, we discuss common problems and questions related to Bayesian networks. We recommend Bayesian networks be investigated to gain causal insight in psychological data. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Network approaches to psychopathology have become increasingly common in mental health research, with many theoretical and methodological developments quickly gaining traction. This article illustrates contemporary practices in applying network analytical tools, bridging the gap between network concepts and their empirical applications. We explain how we can use graphs to construct networks representing complex associations among observable psychological variables. We then discuss key network models, including dynamic networks, time‐varying networks, network models derived from panel data, network intervention analysis, latent networks, and moderated models. In addition, we discuss Bayesian networks and their role in causal inference with a focus on cross‐sectional data. After presenting the different methods, we discuss how network models and psychopathology theories can meaningfully inform each other. We conclude with a discussion that summarizes the insights each technique can provide in mental health research.

Learning the structure of Bayesian networks from data is known to be a computationally challenging, NP-hard problem. The literature has long investigated how to perform structure learning from data containing large numbers of variables, following a general interest in high-dimensional applications (“small n, large p”) in systems biology and genetics. More recently, data sets with large numbers of observations (the so-called “big data”) have become increasingly common; and these data sets are not necessarily high-dimensional, sometimes having only a few tens of variables depending on the application. We revisit the computational complexity of Bayesian network structure learning in this setting, showing that the common choice of measuring it with the number of estimated local distributions leads to unrealistic time complexity estimates for the most common class of score-based algorithms, greedy search. We then derive more accurate expressions under common distributional assumptions. These expressions suggest that the speed of Bayesian network learning can be improved by taking advantage of the availability of closed-form estimators for local distributions with few parents. Furthermore, we find that using predictive instead of in-sample goodness-of-fit scores improves speed; and we confirm that it improves the accuracy of network reconstruction as well, as previously observed by Chickering and Heckerman (Stat Comput 10: 55–62, 2000 ). We demonstrate these results on large real-world environmental and epidemiological data; and on reference data sets available from public repositories.

A classic approach for learning Bayesian networks from data is to identify a maximum a posteriori (MAP) network structure. In the case of discrete Bayesian networks, MAP networks are selected by maximising one of several possible Bayesian–Dirichlet (BD) scores; the most famous is the Bayesian–Dirichlet equivalent uniform (BDeu) score from Heckerman et al. (Mach Learn 20(3):197–243, 1995 ). The key properties of BDeu arise from its uniform prior over the parameters of each local distribution in the network, which makes structure learning computationally efficient; it does not require the elicitation of prior knowledge from experts; and it satisfies score equivalence. In this paper we will review the derivation and the properties of BD scores, and of BDeu in particular, and we will link them to the corresponding entropy estimates to study them from an information theoretic perspective. To this end, we will work in the context of the foundational work of Giffin and Caticha (Proceedings of the 27th international workshop on Bayesian inference and maximum entropy methods in science and engineering, pp 74–84, 2007 ), who showed that Bayesian inference can be framed as a particular case of the maximum relative entropy principle. We will use this connection to show that BDeu should not be used for structure learning from sparse data, since it violates the maximum relative entropy principle; and that it is also problematic from a more classic Bayesian model selection perspective, because it produces Bayes factors that are sensitive to the value of its only hyperparameter. Using a large simulation study, we found in our previous work [Scutari in J Mach Learn Res (Proc Track PGM 2016) 52:438–448, 2016 ] that the Bayesian–Dirichlet sparse (BDs) score seems to provide better accuracy in structure learning; in this paper we further show that BDs does not suffer from the issues above, and we recommend to use it for sparse data instead of BDeu. Finally, will show that these issues are in fact different aspects of the same problem and a consequence of the distributional assumptions of the prior.

Clinical decision making regarding the treatment of unruptured intracranial aneurysms (IA) benefits from a better understanding of the interplay of IA rupture risk factors. Probabilistic graphical models can capture and graphically display potentially causal relationships in a mechanistic model. In this study, Bayesian networks (BN) were used to estimate IA rupture risk factors influences. From 1248 IA patient records, a retrospective, single-cohort, patient-level data set with 9 phenotypic rupture risk factors (n=790 complete entries) was extracted. Prior knowledge together with score-based structure learning algorithms estimated rupture risk factor interactions. Two approaches, discrete and mixed-data additive BN, were implemented and compared. The corresponding graphs were learned using non-parametric bootstrapping and Markov chain Monte Carlo, respectively. The BN models were compared to standard descriptive and regression analysis methods. Correlation and regression analyses showed significant associations between IA rupture status and patient's sex, familial history of IA, age at IA diagnosis, IA location, IA size and IA multiplicity. BN models confirmed the findings from standard analysis methods. More precisely, they directly associated IA rupture with familial history of IA, IA size and IA location in a discrete framework. Additive model formulation, enabling mixed-data, found that IA rupture was directly influenced by patient age at diagnosis besides additional mutual influences of the risk factors. This study establishes a data-driven methodology for mechanistic disease modelling of IA rupture and shows the potential to direct clinical decision-making in IA treatment, allowing personalised prediction.