  Gaussian processes (GPs) have been extensively utilized as nonparametric
models for component separation in 21 cm data analyses. This exploits the
distinct spectral behavior of the cosmological and foreground signals, which
are modeled through the GP covariance kernel. Previous approaches have employed
a global GP kernel along all lines of sight (LoS). In this work, we study
Bayesian approaches that allow for spatial variations in foreground kernel
parameters, testing them against simulated HI intensity mapping observations.
We consider a no-pooling (NP) model, which treats each LoS independently by
fitting for separate covariance kernels, and a hierarchical Gaussian Process
(HGP) model that allows for variation in kernel parameters between different
LoS, regularized through a global hyperprior. We find that accounting for
spatial variations in the GP kernel parameters results in a significant
improvement in cosmological signal recovery, achieving up to a 30% reduction in
the standard deviation of the residual distribution and improved model
predictive performance. Allowing for spatial variations in GP kernel parameters
also improves the recovery of the HI power spectra and wavelet scattering
transform coefficients. Whilst the NP model achieves superior recovery as
measured by the residual distribution, it demands extensive computational
resources, faces formidable convergence challenges, and is prone to
overfitting. Conversely, the HGP model strikes a balance between the accuracy
and robustness of the signal recovery. Further improvements to the HGP model
will require more physically motivated modeling of foreground spatial
variations.
