Advancing multilevel Bayesian networks with efficient Bayesian inference.

Bayesian networks (BNs) are widely used tools for the modeling of complex dependencies among variables. However, their application to multilevel or clustered data remains constrained, especially in scenarios involving multiple random effects, due to a lack of methods and more pressing, efficient computational frameworks. In this article, we propose an innovative framework, INLA-MBN, that integrates multilevel BNs (MBNs) with the integrated nested Laplace approximation (INLA), thereby facilitating efficient structure and parameter learning in both longitudinal and cross-sectional multilevel data contexts. To the best of our knowledge, this is the first implementation of INLA for BNs that involves more than one correlated random effect. More precisely, we advance existing research by modeling longitudinal data with two correlated random effects and multilevel cross-sectional data with more than two correlated random effects. Our approach encompasses both Gaussian and hybrid MBNs that incorporate Gaussian and categorical nodes. Through an exhaustive simulation study, we demonstrate that INLA-MBN exhibit superior performance compared to networks based on maximum likelihood estimation, particularly, in scenarios characterized by small group sizes or limited repeated measurements per subject. INLA-MBN achieves a fully inferred MBN, while negating the cost associated with MBNs based on Markov Chain Monte Carlo. The proposed method was also applied to real-life longitudinal child morbidity data to demonstrate its practical applicability.