Distributed Bayesian Varying Coefficient Modeling Using a Gaussian Process Prior
Distributed Bayesian Varying Coefficient Modeling Using a Gaussian Process Prior
复制标题
DOI：
--
复制
发表时间：
2020-06
期刊：
影响因子：
--
通讯作者：
Rajarshi Guhaniyogi;Cheng Li;T. Savitsky;Sanvesh Srivastava
中科院分区：
文献类型：
--
作者：
Rajarshi Guhaniyogi;Cheng Li;T. Savitsky;Sanvesh Srivastava
变系数模型（VCM）被广泛用于函数数据模型中的非线性回归函数的估计。然而，他们的贝叶斯变体使用高斯过程（GP）先验的函数系数，在大规模数据应用中受到了有限的关注。这主要是由于使用马尔可夫链蒙特卡罗（MCMC）算法的后验计算非常慢。我们使用分而治之的贝叶斯方法，分三步来解决这个问题。第一步通过从完整数据中进行采样而不进行替换，创建大量样本量小得多的数据子集。第二步制定VCM作为一个线性混合效应模型，并开发了一个数据增强（DA）型算法，用于并行获得MCMC绘制的参数和预测的所有子集。DA型算法适当地修改了可能性，使得每个子集后验分布是对应的真实后验分布的精确近似。第三步开发一种组合算法，用于将基于MCMC的子集后验分布估计聚合为单个后验分布，称为聚合蒙特卡罗（AMC）后验。理论上，我们得到了变系数和均值回归函数的AMC后验分布的minimax最优后验收敛速度。我们提供了量化的顺序的子集样本大小和数量的子集，根据光滑性的多元GP。实证结果表明，满足我们的理论假设的组合方案，包括在AMC算法中的一个，有更好的名义覆盖率，更短的可信区间，更小的均方误差，更高的有效样本量比他们的主要竞争对手在不同的模拟和真实的数据分析。
Varying coefficient models (VCMs) are widely used for estimating nonlinear regression functions in functional data models. Their Bayesian variants using Gaussian process (GP) priors on the functional coefficients, however, have received limited attention in massive data applications. This is primarily due to the prohibitively slow posterior computations using Markov chain Monte Carlo (MCMC) algorithms. We address this problem using a divide-and-conquer Bayesian approach that operates in three steps. The first step creates a large number of data subsets with much smaller sample sizes by sampling without replacement from the full data. The second step formulates VCM as a linear mixed-effects model and develops a data augmentation (DA)-type algorithm for obtaining MCMC draws of the parameters and predictions on all the subsets in parallel. The DA-type algorithm appropriately modifies the likelihood such that every subset posterior distribution is an accurate approximation of the corresponding true posterior distribution. The third step develops a combination algorithm for aggregating MCMC-based estimates of the subset posterior distributions into a single posterior distribution called the Aggregated Monte Carlo (AMC) posterior. Theoretically, we derive minimax optimal posterior convergence rates for the AMC posterior distributions of both the varying coefficients and the mean regression function. We provide quantification on the orders of subset sample sizes and the number of subsets according to the smoothness properties of the multivariate GP. The empirical results show that the combination schemes that satisfy our theoretical assumptions, including the one in the AMC algorithm, have better nominal coverage, shorter credible intervals, smaller mean square errors, and higher effective sample size than their main competitors across diverse simulations and in a real data analysis.