We study the use of Gaussian process emulators to approximate the parameter-to-observation map or the negative log-likelihood in Bayesian inverse problems. We prove error bounds on the Hellinger distance between the true posterior distribution and various approximations based on the Gaussian process emulator. Our analysis includes approximations based on the mean of the predictive process, as well as approximations based on the full Gaussian process emulator. Our results show that the Hellinger distance between the true posterior and its approximations can be bounded by moments of the error in the emulator. Numerical results confirm our theoretical findings.

翻译：本研究探讨了在贝叶斯反问题中，使用高斯过程仿真器近似参数-观测映射或负对数似然的方法。我们证明了真实后验分布与基于高斯过程仿真器的各类近似之间的Hellinger距离误差界。我们的分析涵盖了基于预测过程均值的近似方法，以及基于完整高斯过程仿真器的近似方法。研究结果表明，真实后验与其近似之间的Hellinger距离可通过仿真器误差的矩进行界定。数值实验结果验证了我们的理论发现。