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. In contrast, the Laplace approximation is shown 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：提出了一种通过自适应高斯求积在具有单一分组因子的混合模型中进行近似边际似然推断的方法。核心技术贡献在于提出了一种计算近似对数边际似然精确梯度的算法。这使得通过拟牛顿优化实现高效最大似然估计成为可能，实验证明该方法比基于有限差分梯度或无导数优化的现有方法速度更快。该方法专门针对具有多元相关高斯随机效应的伯努利混合模型：此处使用高斯密度的逆对数-Cholesky参数化进行计算，在模型拟合过程中无需进行矩阵分解，同时为原始尺度上的方差参数提供Wald置信区间。模拟实验证明：若使用足够多的求积点，对于由大量极小组构成且组间异质性较大的数据，这些区间能达到名义覆盖水平。相比之下，拉普拉斯近似在由大量小组构成的数据上显示出特别差的覆盖率和较高的偏差。自适应求积缓解了这一问题，本文提出的方法提高了这种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}的代码复现。