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.

翻译：马尔可夫链蒙特卡洛（MCMC）方法仍是贝叶斯结构方程模型（SEM）估计的主流方法，但常伴随高昂的计算成本。我们提出一种针对SEM的定制化近似贝叶斯方法，借鉴集成嵌套拉普拉斯近似（INLA, Rue等, 2009, J. R. Stat. Soc. Series B Stat. Methodol.）框架的思想。我们实现了一种简化拉普拉斯近似，通过在每个参数方向上高效剖分后验密度并校正非对称性，从而对边缘分布进行参数化偏态正态估计。此外，我们应用变分贝叶斯校正来平移边缘位置，从而更准确地捕捉后验质量。因子得分和模型拟合指数等关键量通过调整后的高斯连接函数采样方案获得。对于正态理论SEM，该方法为基于采样的推断提供了高精度替代方案，在保持完整贝叶斯推断精度的同时，实现接近“极大似然”的计算速度。