Approximate Bayesian Inference for Structural Equation Models using Integrated Nested Laplace Approximations

TL;DR

This paper introduces a fast, approximate Bayesian inference method for structural equation models using integrated nested Laplace approximations, offering a computationally efficient alternative to traditional MCMC methods.

Contribution

The authors develop a novel simplified Laplace approximation combined with variational Bayes correction for SEM, enabling rapid and accurate Bayesian inference without extensive sampling.

Findings

Achieves near-maximum likelihood inference speeds

Provides accurate posterior marginals with skew-normal correction

Offers an efficient alternative to MCMC for SEM

Abstract

Markov chain Monte Carlo (MCMC) methods remain the mainstay of Bayesian estimation of structural equation models (SEM), though they often incur a high computational cost. We present a bespoke approximate Bayesian approach to SEM, drawing on ideas from the integrated nested Laplace approximation (INLA, Rue et al., 2009, J. R. Stat. Soc. Series B Stat. Methodol.) framework. We implement a simplified Laplace approximation that efficiently profiles the posterior density in each parameter direction while correcting for asymmetry, allowing for parametric skew-normal estimation of the marginals. Furthermore, we apply a variational Bayes correction to shift the marginal locations, thereby better capturing the posterior mass. Essential quantities, including factor scores and model-fit indices, are obtained via an adjusted Gaussian copula sampling scheme. For normal-theory SEM, this approach offers a highly accurate alternative to sampling-based inference, achieving near-'maximum likelihood' speeds while retaining the precision of full Bayesian inference.