A projection-based Laplace approximation for spatial latent variable models

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.