  Structural equation models (SEMs) are commonly used to study the structural
relationship between observed variables and latent constructs. Recently,
Bayesian fitting procedures for SEMs have received more attention thanks to
their potential to facilitate the adoption of more flexible model structures,
and variational approximations have been shown to provide fast and accurate
inference for Bayesian analysis of SEMs. However, the application of
variational approximations is currently limited to very simple, elemental SEMs.
We develop mean-field variational Bayes algorithms for two SEM formulations for
data that present non-Gaussian features such as skewness and multimodality. The
proposed models exploit the use of mixtures of Gaussians, include covariates
for the analysis of latent traits and consider missing data. We also examine
two variational information criteria for model selection that are
straightforward to compute in our variational inference framework. The
performance of the MFVB algorithms and information criteria is investigated in
a simulated data study and a real data application.
