Nonparametric maximum likelihood component separation

Mitigation of the impact of foreground contributions to measurements of cosmic microwave background (CMB) polarization is a crucial step in modern CMB data analysis and is of particular importance for a detection of large-scale CMB B modes. A large variety of techniques, based on different assumptions and aiming at either a full component separation or merely cleaning the foreground signals from the CMB maps, have been described in the literature. In this work, we consider this problem within a unified framework based on the maximum likelihood principle, under the assumption that the signal at each frequency can be represented as a linear mixture of sky templates. We discuss the impact of various additional assumptions on the final outcome of the procedure. We find that the component separation problem can be fully solved in two specific situations: when we either know the frequency scaling of all the components or can correctly model them with a limited number of unknown parameters, as is the case in the parametric component separation techniques; or when we either know the statistical properties of all the components, the foregrounds and CMB, or can correctly model them with a limited number of parameters, as for instance in the spectral matching independent component analysis (SMICA) approach. However, we also show that much less stringent assumptions are sufficient if we only aim at recovering the cleaned CMB signal. In particular, we discuss a "minimally informed" nonparametric method based on maximum likelihood. The method only assumes that the component properties are independent on the sky direction in some, possibly limited, region of the sky and that the CMB covariance is known up to some limited number of parameters. We apply this method to recover the CMB B modes polarization signal in the context of forthcoming CMB experiments and compare its performance with that of the standard parametric maximum likelihood approach. We show that the minimally informed method performs as well as the parametric one when the parametric model is correct, and supersedes it when the it is incorrect.