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.

中文翻译：

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潜在高斯随机场模型的降维蒙特卡罗最大似然

蒙特卡罗最大似然 (MCML) 提供了一种优雅的方法来寻找潜在变量模型的最大似然估计量 (MLE)。然而，当潜在变量是高维且相关时，MCML 算法的计算成本很高，就像潜在高斯随机场模型的情况一样。潜在高斯随机场模型被广泛使用，例如，在构建灵活的回归模型和空间相关数据的插值等许多研究领域中，例如分析疾病建模中的计数数据和冰盖的存在-不存在卫星图像。我们通过使用基于投影的方法来减少随机效应的维度，提出了一种计算效率高的 MCML 算法。我们开发了一种迭代方法来寻找有效的重要性函数；这通常是一个具有挑战性的问题，并且对于 MCML 算法在计算上可行至关重要。我们发现我们的方法适用于连续（潜在高斯过程）和离散域（潜在高斯马尔可夫随机场）模型。我们说明了我们的方法在具有挑战性的模拟和真实数据示例中的应用，否则最大似然估计将非常具有挑战性。此外，我们研究了潜变量模型的 MCML 方法中一个经常被忽视的挑战：计算结果估计的标准误差的实际问题，以及评估结果置信区间是否提供名义覆盖率。因此，我们的研究为高维潜变量模型实现 MCML 算法的细节提供了有用的见解。