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. Kernel recovery and mean function inference were explored on synthetic data from exoplanet transit light curve simulations. 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 $H_0= 66 \pm 6\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, $H_0= 67 \pm 10\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and $H_0= 69 \pm 6\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, 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：采用全贝叶斯方法，将高斯过程回归扩展为包含核函数选择及核超参数的边缘化。此外，通过证据进行贝叶斯模型比较，实现了核函数的直接对比。联合后验的计算通过一个跨维度采样器实现，该采样器通过在更高维空间中嵌入离散核函数选择及其超参数，并利用嵌套采样从中获取样本，从而同时对其进行采样。针对系外行星凌星光变曲线模拟生成的合成数据，探讨了核函数恢复与均值函数推断。随后，该方法被扩展至均值函数与噪声模型的边缘化，并应用于从哈勃参数随红移变化的实测数据中推断当今哈勃参数$H_0$，这些数据源自宇宙学模型无关的宇宙时钟观测以及依赖$\Lambda$CDM模型的重子声学振荡观测。从宇宙时钟数据、重子声学振荡数据及联合数据集推断的$H_0$值分别为$H_0= 66 \pm 6\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$、$H_0= 67 \pm 10\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$和$H_0= 69 \pm 6\, \mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$。宇宙时钟数据集的核函数后验更倾向于非平稳线性核函数。最后，研究显示这些数据集与$\ln R=12.17\pm 0.02$不存在矛盾。