We present in this paper a new Bayesian semi-blind approach for foreground removal in observations of the 21-cm signal with interferometers. The technique, which we call HIEMICA (HI Expectation-Maximization Independent Component Analysis), is an extension of the Independent Component Analysis (ICA) technique developed for two-dimensional (2D) CMB maps to three-dimensional (3D) 21-cm cosmological signals measured by interferometers. This technique provides a fully Bayesian inference of power spectra and maps and separates the foregrounds from signal based on the diversity of their power spectra. Only relying on the statistical independence of the components, this approach can jointly estimate the 3D power spectrum of the 21-cm signal and, the 2D angular power spectrum and the frequency dependence of each foreground component, without any prior assumptions about foregrounds. This approach has been tested extensively by applying it to mock data from interferometric 21-cm intensity mapping observations under idealized assumptions of instrumental effects. We also discuss the impact when the noise properties are not known completely. As a first step toward solving the 21 cm power spectrum analysis problem we compare the semi-blind HIEMICA technique with the commonly used Principal Component Analysis (PCA). Under the same idealized circumstances the proposed technique provides significantly improved recovery of the power spectrum. This technique can be applied straightforwardly to all 21-cm interferometric observations, including epoch of reionization measurements, and can be extended to single-dish observations as well.
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