Bayes-SCF: A Bayesian filter to mitigate foreground leakage in the 21-cm power spectrum

Missing channels in radio-interferometric visibility data can introduce systematic artifacts into the estimated 21-cm power spectrum. A common workaround is to first estimate the two-frequency correlation $C(\Delta\nu)$ and then Fourier-transform it to obtain the power spectrum $P(k_\parallel)$. This procedure yields an unbiased estimate when the signal is statistically homogeneous (ergodic) along the line-of-sight, but it fails in the presence of non-ergodic foregrounds. Smooth Component Filtering (SCF) has recently been proposed as a solution to this problem, in which the dominant non-ergodic (spectrally smooth) component is removed prior to estimating $C(\Delta\nu)$. In existing implementations, the smooth component is estimated by convolving the visibilities with a Hann window along the frequency axis. We demonstrate that this Hann-based SCF performs adequately only when foregrounds are extremely spectrally smooth, i.e., when they possess a long frequency-correlation length. In contrast, it breaks down when foregrounds exhibit short correlation lengths, as is frequently encountered in real observations. We introduce a Bayesian extension, Bayes-SCF, based on Gaussian Process (GP) regression, which overcomes this limitation. Bayes-SCF models the smooth component via a covariance function with a fixed correlation length, enabling a controlled and data-driven filtering. Using simulated data, we show that Bayes-SCF robustly recovers the input model 21-cm power spectrum even in the presence of spectrally unsmooth foregrounds. Bayes-SCF is also effective in a delay-spectrum approach. The primary trade-off introduced by the Bayesian framework is increased computational cost; future work will focus on optimizing the algorithm and applying it to real MWA data.