Using a fully Bayesian approach, Gaussian Process regression is extended to include marginalisation over the kernel choice and kernel hyperparameters. In addition, Bayesian model comparison via the evidence enables direct kernel comparison. The calculation of the joint posterior was implemented with a transdimensional sampler which simultaneously samples over the discrete kernel choice and their hyperparameters by embedding these in a higher-dimensional space, from which samples are taken using nested sampling. This method was explored on synthetic data from exoplanet transit light curve simulations. The true kernel was recovered in the low noise region while no kernel was preferred for larger noise. Furthermore, inference of the physical exoplanet hyperparameters was conducted. In the high noise region, either the bias in the posteriors was removed, the posteriors were broadened or the accuracy of the inference was increased. In addition, the uncertainty in mean function predictive distribution increased due to the uncertainty in the kernel choice. Subsequently, the method was extended to marginalisation over mean functions and noise models and applied to the inference of the present-day Hubble parameter, $H_0$, from real measurements of the Hubble parameter as a function of redshift, derived from the cosmologically model-independent cosmic chronometer and {\Lambda}CDM-dependent baryon acoustic oscillation observations. The inferred $H_0$ values from the cosmic chronometers, baryon acoustic oscillations and combined datasets are $H_0$ = 66$\pm$6 km/s/Mpc, $H_0$ = 67$\pm$10 km/s/Mpc and $H_0$ = 69$\pm$6 km/s/Mpc, respectively. The kernel posterior of the cosmic chronometers dataset prefers a non-stationary linear kernel. Finally, the datasets are shown to be not in tension with ln(R)=12.17$\pm$0.02.

翻译：采用完全贝叶斯方法，将高斯过程回归扩展至包含核选择与核超参数的边缘化。此外，通过证据进行贝叶斯模型比较，实现了核的直接对比。联合后验计算通过一种跨维度采样器实现，该采样器将离散核选择及其超参数嵌入高维空间后，利用嵌套采样同步采样。该方法在系外行星凌星光变曲线模拟生成的合成数据上进行了验证：在低噪声区域可恢复真实核，而高噪声区域无特定核被优先选择。进一步开展了物理系外行星超参数的推断，在高噪声区域中，后验偏差被消除、后验分布展宽或推断精度得以提升。此外，由于核选择的不确定性，均值函数预测分布的不确定性亦有所增加。随后，该方法被扩展至均值函数与噪声模型的边缘化，并应用于从真实哈勃参数（作为红移函数）推断当前哈勃参数$H_0$的实测数据中，这些数据源自宇宙学模型无关的宇宙计时器与依赖ΛCDM的重子声学振荡观测。由宇宙计时器、重子声学振荡及联合数据集推断的$H_0$值分别为$H_0$ = 66±6 km/s/Mpc、$H_0$ = 67±10 km/s/Mpc和$H_0$ = 69±6 km/s/Mpc。宇宙计时器数据集的核后验倾向于非平稳线性核。最后，数据集与ln(R)=12.17±0.02未呈现紧张关系。