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

Haziq Jamil, Håvard Rue

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.

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

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.

Paper Structure

This paper contains 24 sections, 3 theorems, 45 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Laplace's Method
  4. Skew-Normal Distribution
  5. Methodology
  6. Joint Laplace Approximation
  7. Skew-Normal Laplace Profiling of Posterior Marginals
  8. Conditional Mean Path
  9. Efficient Volume Correction
  10. Skew-Normal Curve Fitting
  11. Variational Mean Correction of the Laplace Approximation
  12. Gaussian Copula Sampling
  13. Validation Study
  14. Benchmark Model and Priors
  15. Computational Note
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let $\boldsymbol\vartheta\sim\mathrm{N}_m(\boldsymbol\vartheta^*,\boldsymbol\Omega)$ with $\boldsymbol\Omega \succ \mathbf 0$. Then the marginal density of the $j$th component, $j=1,\dots,m$, is proportional to the joint density evaluated along its conditional mean path:

Figures (5)

  • Figure 1: Path diagram for the Political Democracy SEM bollen1989structural. Circles denote latent variables; squares denote observed indicators. Single-headed arrows represent factor loadings ($\Lambda_{i,j}$) and structural regression coefficients ($B_{j,j'}$). Double-headed arrows represent residual variances ($\Theta_{i,i}$), residual covariances ($\Theta_{i,j}$, $i \neq j$), and latent variances ($\Psi_{j,j}$). The diagram depicts the model for a single observation; the same structure is assumed independently for each $s = 1, \ldots, n$.
  • Figure 2: Marginal posterior densities for the Political Democracy SEM. Solid curves show the fitted skew-normal approximation; shaded densities are kernel density estimates from MCMC. Percentages indicate JS similarity, with values near 100% reflecting close distributional agreement.
  • Figure 3: $B=250$ fitted marginal densities superimposed for each parameter/sample-size combination, with the true value marked by a vertical dashed line and summary metrics annotated in each panel.
  • Figure 4: PIT-ECDF diagnostic plots for all 42 parameters under diffuse and informative priors. The black diagonal line indicates the expected ECDF under perfect calibration, while the shaded grey area represents 95% simultaneous confidence bands derived from the exact distribution of uniform order statistics. Deviations of the ECDF curves outside the confidence bands indicate miscalibration.
  • Figure 5: Quantile-quantile plots of true parameter values for successful and failed SBC replications under diffuse priors. The vertical dashed line indicates the median of the nominal prior distribution. Successful fits cluster around moderate parameter values, while failed fits are associated with extreme configurations, particularly for the parameters that exhibit miscalibration in Figure \ref{['fig-sbc-ecdf']}.

Theorems & Definitions (3)

  • Lemma 3.1: Marginalisation via the CMP
  • Lemma 3.2: Volume-slope decomposition
  • Corollary 3.1: Gradient representation of the volume slope