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）方法仍是结构方程模型（SEM）贝叶斯估计的主流手段，但往往伴随着高昂的计算成本。我们基于集成嵌套拉普拉斯逼近（INLA, Rue 等, 2009, J. R. Stat. Soc. Series B Stat. Methodol.）框架的思想，提出一种面向SEM的定制化近似贝叶斯方法。通过实现简化的拉普拉斯逼近，我们能够在每个参数方向上高效地轮廓化后验密度并校正非对称性，从而实现对边际密度进行参数化偏正态估计。在此基础上，应用变分贝叶斯校正移动边际位置，以更准确地捕捉后验质量。通过调整高斯Copula抽样方案，可获取因子得分、模型拟合指数等关键量。对于正态理论SEM，该方法为基于抽样的推断提供了高精度替代方案，在保持完全贝叶斯推断精度的同时，实现了近乎"极大似然"级别的计算速度。