Fitting Structural Equation Models via Variational Approximations

Structural equation models are commonly used to capture the structural relationship between sets of observed and unobservable variables. In Bayesian settings, fitting and inference for these models are typically performed via Markov chain Monte Carlo methods that may be computationally intensive, especially for models with a large number of manifest variables or complex structures. Variational approximations can be a fast alternative; however they have not been adequately explored for this class of models. We develop a mean field variational Bayes approach for fitting basic structural equation models. We show that this variational approximation method can provide reliable inference while being significantly faster than Markov chain Monte Carlo. Classical mean field variational Bayes may underestimate the true posterior variance, therefore we propose and study bootstrap to overcome this issue. We discuss different inference strategies based on bootstrap and demonstrate how these can considerably improve the accuracy of the variational approximation through real and simulated examples.