A method is introduced for approximate marginal likelihood inference via adaptive Gaussian quadrature in mixed models with a single grouping factor. The core technical contribution is an algorithm for computing the exact gradient of the approximate log-marginal likelihood. This leads to efficient maximum likelihood via quasi-Newton optimization that is demonstrated to be faster than existing approaches based on finite-differenced gradients or derivative-free optimization. The method is specialized to Bernoulli mixed models with multivariate, correlated Gaussian random effects; here computations are performed using an inverse log-Cholesky parameterization of the Gaussian density that involves no matrix decomposition during model fitting, while Wald confidence intervals are provided for variance parameters on the original scale. Simulations give evidence of these intervals attaining nominal coverage if enough quadrature points are used, for data comprised of a large number of very small groups exhibiting large between-group heterogeneity. The Laplace approximation is well-known to give especially poor coverage and high bias for data comprised of a large number of small groups. Adaptive quadrature mitigates this, and the methods in this paper improve the computational feasibility of this more accurate method. All results may be reproduced using code available at \url{https://github.com/awstringer1/aghmm-paper-code}.

翻译：本文提出了一种针对单一分组因子混合模型，通过自适应高斯求积进行近似边际似然推断的方法。其核心技术创新在于提出了一种计算近似对数边际似然精确梯度的算法。该算法通过拟牛顿优化实现了高效的最大似然估计，并被证明比基于有限差分梯度或无导数优化的现有方法更为快速。该方法特别适用于具有多元相关高斯随机效应的伯努利混合模型；在此，计算采用高斯密度的逆对数-Cholesky参数化进行，该参数化在模型拟合过程中无需矩阵分解，同时为原始尺度上的方差参数提供了Wald置信区间。模拟结果表明，对于由大量规模极小且组间异质性强的组构成的数据，若使用足够多的求积点，这些区间能够达到名义覆盖水平。众所周知，拉普拉斯近似对于由大量小规模组构成的数据，其覆盖率和偏差表现尤其不佳。自适应求积法则能缓解此问题，而本文提出的方法提升了这一更精确方法的计算可行性。所有结果均可通过\url{https://github.com/awstringer1/aghmm-paper-code}提供的代码复现。