We investigate Bayesian variable selection in models driven by Gaussian processes, which allows us to treat linear, nonlinear and nonparametric models, in conjunction with even dependent setups, in the same vein. We consider the Bayes factor route to variable selection, and develop a general asymptotic theory for the Gaussian process framework in the “large p, large n” settings even with \(p\gg n\), establishing almost sure exponential convergence of the Bayes factor under appropriately mild conditions. The fixed p setup is included as a special case. To illustrate, we apply our result to variable selection in linear regression, Gaussian process model with squared exponential covariance function accommodating the covariates, and a first-order autoregressive process with time-varying covariates. We also follow up our theoretical investigations with ample simulation experiments in the above regression contexts and variable selection in a real, riboflavin data consisting of 71 observations and 4088 covariates. For implementation of variable selection using Bayes factors, we develop a novel and effective general-purpose transdimensional, transformation-based Markov chain Monte Carlo algorithm, which has played a crucial role in simulated and real data applications.

我们研究了由高斯过程驱动的模型中的贝叶斯变量选择，这使我们能够以同样的方式处理线性、非线性和非参数模型，甚至与相关设置相结合。我们考虑了变量选择的贝叶斯因子路径，并在“大p , 大n ”设置下为高斯过程框架开发了一般渐近理论，即使使用\(p\gg n\)，建立几乎可以肯定的贝叶斯指数收敛因素在适当温和的条件下。固定p设置作为特殊情况包括在内。为了说明这一点，我们将我们的结果应用于线性回归中的变量选择、具有适应协变量的平方指数协方差函数的高斯过程模型以及具有时变协变量的一阶自回归过程。我们还在上述回归背景下进行了大量模拟实验，并在由 71 个观察值和 4088 个协变量组成的真实核黄素数据中进行了变量选择，以此来跟进我们的理论研究。为了使用贝叶斯因子实现变量选择，我们开发了一种新颖有效的通用跨维、基于变换的马尔可夫链蒙特卡罗算法，该算法在模拟和真实数据应用中发挥了至关重要的作用。