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

Haziq Jamil; Håvard Rue

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arXiv · 2603.25690

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

Haziq Jamil·Håvard Rue

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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., B: Stat. Methodol.) framework. After analytically integrating out the latent variables, we implement a simplified Laplace approximation that profiles the posterior using an efficient gradient-based volume correction, allowing for parametric skew-normal estimation of the marginals. A variational Bayes correction further refines the centre of the joint Gaussian approximation; marginal profiling implicitly recovers the same location correction to leading order. Essential quantities, including factor scores and model-fit indices, are obtained via a Gaussian copula sampling scheme. On a benchmark SEM, inference completes in seconds with close agreement to MCMC. A simulation study stresses the approximation along sample size, identification, boundary proximity, and model dimension. Accuracy remains high across the examined sample sizes and model dimensions, with degradation concentrated in weakly identified settings and in the lower tails of variances near the zero boundary.

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Haziq Jamil, Håvard Rue. 2026-09-16. Approximate Bayesian Inference for Structural Equation Models using Integrated Nested Laplace Approximations. https://arxiv.org/abs/2603.25690

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