  In this article we present a formalism to incorporate the partial-sky maps to
the Gibbs ILC algorithm to estimate the joint posterior density of the Cosmic
Microwave Background (CMB) signal and the theoretical CMB angular power
spectrum given the observed CMB maps. In order to generate the partial-sky maps
we mask all the observed CMB maps provided by WMAP and Planck satellite mission
using a Gaussian smoothed mask formed on the basis of thermal dust emissions in
Planck 353 GHz map. The central galactic region from all the input maps is
removed after the application of the smoothed mask. While implementing the
Gibbs ILC method on the partial-sky maps, we convert the partial-sky cleaned
angular power spectrum to the full-sky angular power spectrum using the
mode-mode coupling matrix estimated from the smoothed mask. The main products
of our analysis are partial-sky cleaned best-fit CMB map and an estimate of the
underlying full-sky theoretical CMB angular power spectrum along with their
error estimates. We validate the methodology by performing detailed Monte Carlo
simulations after using realistic models of foregrounds and detector noise
consistent with the WMAP and Planck frequency channels used in our analysis. We
can estimate the posterior density and full-sky theoretical CMB angular power
spectrum, without any need to explicitly model the foreground components, from
partial-sky maps using our method. Another important feature of this method is
that the power spectrum results along with the error estimates can be directly
used for cosmological parameter estimations.
