Implementation and Workflows for INLA-Based Approximate Bayesian Structural Equation Modelling

Abstract

Bayesian structural equation modelling (BSEM) offers many advantages such as principled uncertainty quantification, small-sample regularisation, and flexible model specification. However, the Markov chain Monte Carlo (MCMC) methods on which it relies are computationally prohibitive for the iterative cycle of specification, criticism, and refinement that careful psychometric practice demands. We present INLAvaan, an R package for fast, approximate Bayesian SEM built around the Integrated Nested Laplace Approximation (INLA) framework for structural equation models developed by Jamil & Rue (2026, arXiv:2603.25690 [stat.ME]). This paper serves as a companion manuscript that describes the architectural decisions and computational strategies underlying the package. Two substantive applications -- a 256-parameter bifactor circumplex model and a multilevel mediation model with full-information missing-data handling -- demonstrate the approach on specifications where MCMC would require hours of run time and careful convergence work. In constrast, INLAvaan delivers calibrated posterior summaries in seconds.

Figures (13)

• Figure 1: The INLAvaan pipeline: initialisation, joint posterior approximation, marginal profiling, and Gaussian copula sampling, yielding a fully fitted Bayesian SEM object.
• Figure 2: A clear improvement in approximation quality of the posterior marginals is seen due to the VB mean-shift correction (VBC) on three selected parameters from the political democracy example: a factor loading (ind60= x3), a residual variance (x3 x3), and a latent variance (dem60 dem60). Percentages are Jensen-Shannon similarities to MCMC (higher is better).
• Figure 3: Fifty draws from $\mathrm{N}(\mathbf{0}, \mathbf{I}_2)$ using pseudo-random sampling (left) and a scrambled Owen-Sobol sequence mapped through $\Phi^{-1}$ (right), overlaid on bivariate normal contours. The Sobol points cover the support more uniformly, reducing Monte Carlo variance in the QMC objective (Equation \ref{['eq-vb-elbo']}).
• Figure 4: Log-profile (left) and corresponding density (right) for the residual variance y1 y1 in the political democracy model. The uncorrected raw log-profile is systematically too wide; the volume correction tilts it to better match the true log-marginal shape. The skew-normal fit (red dashed) closely tracks the corrected log-profile, but may deviate in the tails where the profile is less informative.
• Figure 5: The Inventory of Interpersonal Problems (IIP) circumplex. Left: the eight octants at their ideal equi-spaced angles on the interpersonal circle, with dashed lines marking the Love (horizontal) and Dominance (vertical) axes. Right: a sample item measuring each octant on a 0--4 Likert scale.