Bayesian Regularized SEM: Current Capabilities and Constraints

An important challenge in statistical modeling is to balance how well our model explains the phenomenon under investigation with the parsimony of this explanation. In structural equation modeling (SEM), penalization approaches that add a penalty term to the estimation procedure have been proposed to achieve this balance. An alternative to the classical penalization approach is Bayesian regularized SEM in which the prior distribution serves as the penalty function. Many different shrinkage priors exist, enabling great flexibility in terms of shrinkage behavior. As a result, different types of shrinkage priors have been proposed for use in a wide variety of SEMs. However, the lack of a general framework and the technical details of these shrinkage methods can make it difficult for researchers outside the field of (Bayesian) regularized SEM to understand and apply these methods in their own work. Therefore, the aim of this paper is to provide an overview of Bayesian regularized SEM, with a focus on the types of SEMs in which Bayesian regularization has been applied as well as available software implementations. Through an empirical example, various open-source software packages for (Bayesian) regularized SEM are illustrated and all code is made available online to aid researchers in applying these methods. Finally, reviewing the current capabilities and constraints of Bayesian regularized SEM identifies several directions for future research.