Bayesian model comparison and validation with Gaussian Process Regression for interferometric 21-cm signal recovery

The 21-cm signal from neutral hydrogen is anticipated to reveal critical insights into the formation of early cosmic structures during the Cosmic Dawn and the subsequent Epoch of Reionization. However, the intrinsic faintness of the signal, as opposed to astrophysical foregrounds, poses a formidable challenge for its detection. Motivated by the recent success of machine learning based Gaussian Process Regression (GPR) methods in LOFAR and NenuFAR observations, we perform a Bayesian comparison among five GPR models to account for the simulated 4-hour tracking observations with the SKA-Low telescope. The simulated sky is convolved with the instrumental beam response and includes realistic radio sources and thermal noise from 122 to 134 MHz. A Bayesian model evaluation framework is applied to five GPR models to discern the most effective modelling strategy and determine the optimal model parameters. The GPR model with wedge parametrization ($\textit{Wedge}$) and its extension ($\alpha\textit{Noise}$) with noise scaling achieve the highest Bayesian evidence of the observed data and the least biased 21-cm power spectrum recovery. The $\alpha\textit{Noise}$ and $\textit{Wedge}$ models also forecast the best local power-spectrum recovery, demonstrating fractional differences of $-0.14\%$ and $0.47\%$ respectively, compared to the injected 21-cm power at $k = 0.32\ \mathrm{h\ cMpc}^{-1}$. We additionally perform Bayesian null tests to validate the five models, finding that the two optimal models also pass with the remaining three models yielding spurious detections in data containing no 21-cm signal.