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Using a fully Bayesian approach, Gaussian Process regression is extended toinclude marginalisation over the kernel choice and kernel hyperparameters. Inaddition, Bayesian model comparison via the evidence enables direct kernelcomparison. The calculation of the joint posterior was implemented with atransdimensional sampler which simultaneously samples over the discrete kernelchoice and their hyperparameters by embedding these in a higher-dimensionalspace, from which samples are taken using nested sampling. This method wasexplored on synthetic data from exoplanet transit light curve simulations. Thetrue kernel was recovered in the low noise region while no kernel was preferredfor larger noise. Furthermore, inference of the physical exoplanethyperparameters was conducted. In the high noise region, either the bias in theposteriors was removed, the posteriors were broadened or the accuracy of theinference was increased. In addition, the uncertainty in mean functionpredictive distribution increased due to the uncertainty in the kernel choice.Subsequently, the method was extended to marginalisation over mean functionsand noise models and applied to the inference of the present-day Hubbleparameter, $H_0$, from real measurements of the Hubble parameter as a functionof redshift, derived from the cosmologically model-independent cosmicchronometer and {\Lambda}CDM-dependent baryon acoustic oscillationobservations. The inferred $H_0$ values from the cosmic chronometers, baryonacoustic 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$6 km/s/Mpc, respectively. Thekernel posterior of the cosmic chronometers dataset prefers a non-stationarylinear kernel. Finally, the datasets are shown to be not in tension withln(R)=12.17$\pm$0.02.

Kernel-, mean- and noise-marginalised Gaussian processes for exoplanet transits and $H_0$ inference