Laplace method is a practical tool for obtaining maximum likelihood estimators for a wide class of latent variable models. The main idea is to approximate the integrand using a Gaussian distribution. However, with increasing observations, the Laplace approximation becomes infeasible because the dimension of the correlated latent variables grows, which results in the high-dimensional optimization problem. One important example is spatial latent variable models, which are widely used in many fields, such as ecology, epidemiology, and sociology. Spatial latent variable models are useful for investigating the relationship between spatial covariates or predicting the unobserved area. Here, we propose a fast Laplace approximation based on the dimension reduction of the latent variables. Our methods are faster and have fewer components to be tuned than simulation-based methods such as Markov chain Monte Carlo maximum likelihood and Monte Carlo expectation-maximization. Our approach can be applied to the large non-Gaussian spatial data sets, commonly used in modern environmental sciences. Especially, we show how we may understand spatial patterns of non-Gaussian responses for two case studies: confirmed COVID-19 cases in the United States and thickness of the Antarctic ice sheet. Through simulation studies under different scenarios, we investigate that our method can provide accurate maximum likelihood estimations and predictions quickly. Our study can be widely applicable for practical maximum likelihood inference for high-dimensional random effect models. We provide a freely available R-package that can implement the proposed method.

中文翻译：

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空间潜变量模型的基于投影的拉普拉斯近似

拉普拉斯方法是一种实用的工具，用于获得广泛的潜变量模型的最大似然估计量。主要思想是使用高斯分布来近似被积函数。然而，随着观测值的增加，拉普拉斯近似变得不可行，因为相关潜在变量的维数增加，这导致了高维优化问题。一个重要的例子是空间潜变量模型，它被广泛应用于生态学、流行病学和社会学等许多领域。空间潜变量模型可用于研究空间协变量之间的关系或预测未观察到的区域。在这里，我们提出了一种基于潜在变量降维的快速拉普拉斯近似。与基于模拟的方法（例如马尔可夫链蒙特卡罗最大似然和蒙特卡罗期望最大化）相比，我们的方法更快且需要调整的组件更少。我们的方法可以应用于现代环境科学中常用的大型非高斯空间数据集。特别是，我们展示了我们如何理解两个案例研究的非高斯响应的空间模式：美国确诊的 COVID-19 病例和南极冰盖的厚度。通过不同场景下的模拟研究，我们研究我们的方法可以快速提供准确的最大似然估计和预测。我们的研究可以广泛应用于高维随机效应模型的实际最大似然推断。