Approximate Bayesian inference for structural equation models

Overview

Structural equation models (SEM) are widely used to study causal pathways, latent constructs, and measurement error. Full Bayesian estimation via Markov chain Monte Carlo (MCMC), however, is often too slow for the complexity of modern applications. An approximate Bayesian approach to SEM is presented, drawing on ideas from the integrated nested Laplace approximation (INLA) framework. A Laplace approximation to the joint posterior is computed first, and its mean is then shifted by a variational Bayes correction to better capture the posterior mass. Each marginal is estimated by a simplified Laplace approximation, which profiles the posterior density efficiently along each parameter direction while correcting for asymmetry, yielding a parametric skew-normal fit. An efficient Gaussian copula sampling scheme then delivers the essential quantities: factor scores, model-fit indices, and credible intervals for nonlinear derived parameters such as indirect effects. The approach achieves speeds close to maximum likelihood estimation, while retaining the inferential richness of full Bayesian analysis. The methodology is implemented in the R package INLAvaan, and its speed and accuracy are illustrated against MCMC benchmarks on simulated and real data.