Reduced-Dimensional Monte Carlo Maximum Likelihood for Latent Gaussian Random Field Models

Monte Carlo maximum likelihood (MCML) provides an elegant approach to find maximum likelihood estimators (MLEs) for latent variable models. However, MCML algorithms are computationally expensive when the latent variables are high-dimensional and correlated, as is the case for latent Gaussian random field models. Latent Gaussian random field models are widely used, for example, in building flexible regression models and in the interpolation of spatially dependent data in many research areas such as analyzing count data in disease modeling and presence-absence satellite images of ice sheets. We propose a computationally efficient MCML algorithm by using a projection-based approach to reduce the dimensions of the random effects. We develop an iterative method for finding an effective importance function; this is generally a challenging problem and is crucial for the MCML algorithm to be computationally feasible. We find that our method is applicable to both continuous (latent Gaussian process) and discrete domain (latent Gaussian Markov random field) models. We illustrate the application of our methods to challenging simulated and real data examples for which maximum likelihood estimation would otherwise be very challenging. Furthermore, we study an often overlooked challenge in MCML approaches to latent variable models: practical issues in calculating standard errors of the resulting estimates, and assessing whether resulting confidence intervals provide nominal coverage. Our study therefore provides useful insights into the details of implementing MCML algorithms for high-dimensional latent variable models. Supplementary materials for this article are available online.

蒙特卡洛极大似然（Monte Carlo Maximum Likelihood, MCML）为潜变量模型的极大似然估计量（maximum likelihood estimators, MLEs）求解提供了一种优雅的途径。不过，当潜变量为高维且存在相关性时，MCML算法的计算开销会变得极为高昂——隐高斯随机场模型（latent Gaussian random field models）正是这类场景的典型代表。隐高斯随机场模型应用广泛，例如用于构建灵活的回归模型，以及在诸多研究领域中对空间依赖数据进行插值，比如疾病建模中的计数数据分析、冰盖卫星影像的存在-缺失数据处理。本文提出一种计算高效的MCML算法，通过基于投影的方法降低随机效应的维度。我们开发了一种用于求解有效重要性函数的迭代方法——这一问题通常极具挑战性，同时也是保障MCML算法具备计算可行性的核心关键。我们发现所提方法可同时适用于连续域（隐高斯过程，latent Gaussian process）与离散域（隐高斯马尔可夫随机场，latent Gaussian Markov random field）模型。我们通过若干兼具挑战性的模拟数据与真实数据示例演示了所提方法的应用，这类场景下的极大似然估计原本极具难度。此外，我们还研究了潜变量模型的MCML方法中一个常被忽视的挑战：计算所得估计量的标准误的实际计算问题，以及评估所得置信区间是否具备名义覆盖率。因此，本研究为高维潜变量模型的MCML算法实现细节提供了极具价值的参考。本文的补充材料可在线获取。