  We present a new, semi-analytic framework for estimating the level of
residuals present in CMB maps derived from multi-frequency Cosmic Microwave
Background (CMB) data and forecasting their impact on cosmological parameters.
The data are assumed to contain non-negligible signals of astrophysical and/or
Galactic origin, which we clean using parametric component separation
technique. We account for discrepancies between the foreground model assumed
during the separation procedure and the true one, allowing for differences in
scaling laws and/or their spatial variations. Our estimates and their
uncertainties include both systematic and statistical effects and are averaged
over the instrumental noise and CMB signal realizations. The framework can be
further extended to account self-consistently for existing uncertainties in the
foreground models. We demonstrate and validate the framework on simple study
cases which aim at estimating the tensor-to-scalar ratio, r. The proposed
approach is computationally efficient permitting an investigation of hundreds
of set-ups and foreground models on a single CPU.
