ABSTRACT

We define a novel class of additive models, called Extended Latent Gaussian Models, that allow for a wide range of response distributions and flexible relationships between the additive predictor and mean response. The new class covers a broad range of interesting models including multi-resolution spatial processes, partial likelihood-based survival models, and multivariate measurement error models. Because computation of the exact posterior distribution is infeasible, we develop a fast, scalable approximate Bayesian inference methodology for this class based on nested Gaussian, Laplace, and adaptive quadrature approximations. We prove that the error in these approximate posteriors is $o_p(1)$ under standard conditions, and provide numerical evidence suggesting that our method runs faster and scales to larger datasets than methods based on Integrated Nested Laplace Approximations and Markov chain Monte Carlo, with comparable accuracy. We apply the new method to the mapping of malaria incidence rates in continuous space using aggregated data, mapping leukemia survival hazards using a Cox Proportional-Hazards model with a continuously-varying spatial process, and estimating the mass of the Milky Way Galaxy using noisy multivariate measurements of the positions and velocities of star clusters in its orbit. Supplementary materials for this article are available online.

摘要

我们定义了一类新的加性模型，称为扩展潜在高斯模型，它允许广泛的响应分布和加性预测变量与平均响应之间的灵活关系。新课程涵盖了广泛的有趣模型，包括多分辨率空间过程、基于部分似然的生存模型和多元测量误差模型。由于精确后验分布的计算是不可行的，我们基于嵌套高斯、拉普拉斯和自适应正交近似为此类开发了一种快速、可扩展的近似贝叶斯推理方法。我们证明这些近似后验的误差是$o_p(\mathrm{1个})$在标准条件下，并提供数值证据表明我们的方法比基于集成嵌套拉普拉斯近似和马尔可夫链蒙特卡洛的方法运行得更快并且可以扩展到更大的数据集，具有相当的准确性。我们将新方法应用于使用聚合数据绘制连续空间中的疟疾发病率图，使用具有连续变化空间过程的 Cox 比例风险模型绘制白血病生存危害图，并使用嘈杂的多变量估计银河系的质量测量星团在其轨道上的位置和速度。本文的补充材料可在线获取。