Challenges arising when using Gaussian Process priors in probabilistic modeling are how to carry out exact Bayesian inference and how to account for uncertainty on model parameters when making model-based predictions on out-of-sample data. Using probit regression as a working example, this paper presents a simple methodology based on Markov chain Monte Carlo and the pseudo-marginal approach that efficiently addresses both those questions. The results presented in this paper show improvements over previous approaches for sampling from the posterior distribution of the parameters of the covariance function of the Gaussian Process prior. This is particularly important as it offers a powerful tool to carry out the fully Bayesian treatment of Gaussian Process based statistical models. The results also demonstrate that Monte Carlo based integration of all model parameters is actually feasible providing a superior quantification of uncertainty in predictions. Comparison with respect to state-of-the-art probabilistic classifiers confirm this assertion. Finally, this paper demonstrates with an application on a financial time series that the proposed methodology can exploit sparsity in the inverse covariance of the Gaussian Process prior leading to a computationally efficient Markov chain Monte Carlo approach.

本文提出了一种基于伪边缘法的马尔可夫链蒙特卡罗通用策略，有效地解决了使用高斯过程先验的精确贝叶斯推理和模型参数不确定性导致的数据预测问题。此外，Monte Carlo积分可用于预测中的所有模型参数，这在高斯过程的层次统计模型中具有重要意义，并与最先进的概率分类器进行了广泛的比较。

高斯过程的伪边缘贝叶斯推断