论文标题
高斯工艺模型中的贝叶斯固定域渐进参数
Bayesian Fixed-domain Asymptotics for Covariance Parameters in a Gaussian Process Model
论文作者
论文摘要
高斯过程模型通常包含协方差函数中的有限维度参数，需要从数据中估算。我们研究了具有各向同性母乳协方差函数的通用kriging模型中的协方差参数的贝叶斯固定域渐进剂，该协方差在空间统计中有许多应用。我们表明，当域的尺寸小于或等于三个时，可以将微连接参数的关节后验分布和范围参数独立计入固定剂渐近域下的边缘后期乘积。微能参数的后部渐近地在总变化距离上与正态分布的差异差异，而范围参数的后验分布通常不会收敛到任何点质量分布。我们的理论允许对采样点的范围参数和灵活设计的无限支持。我们进一步研究了贝叶斯kriging预测器的后验预测中的渐近效率和收敛速率，其协方差参数从其后验分布中随机得出。在一维的Ornstein-Uhlenbeck过程的特殊情况下，我们明确地得出了范围参数的限制后验和后预测中渐近效率的后收敛速率。我们在数值实验中验证了这些渐近结果。
Gaussian process models typically contain finite dimensional parameters in the covariance function that need to be estimated from the data. We study the Bayesian fixed-domain asymptotics for the covariance parameters in a universal kriging model with an isotropic Matern covariance function, which has many applications in spatial statistics. We show that when the dimension of domain is less than or equal to three, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors under fixed-domain asymptotics. The posterior of the microergodic parameter is asymptotically close in total variation distance to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not converge to any point mass distribution in general. Our theory allows an unbounded prior support for the range parameter and flexible designs of sampling points. We further study the asymptotic efficiency and convergence rates in posterior prediction for the Bayesian kriging predictor with covariance parameters randomly drawn from their posterior distribution. In the special case of one-dimensional Ornstein-Uhlenbeck process, we derive explicitly the limiting posterior of the range parameter and the posterior convergence rate for asymptotic efficiency in posterior prediction. We verify these asymptotic results in numerical experiments.
