  The estimation of the polarization $P$ of extragalactic compact sources in
Cosmic Microwave Background images is a very important task in order to clean
these images for cosmological purposes -- as, for example, to constrain the
tensor-to-scalar ratio of primordial fluctuations during inflation -- and also
to obtain relevant astrophysical information about the compact sources
themselves in a frequency range, $\nu \sim 10$--$200$ GHz, where observations
have only very recently started to be available. In this paper we propose a
Bayesian maximum a posteriori (MAP) approach estimation scheme which
incorporates prior information about the distribution of the polarization
fraction of extragalactic compact sources between 1 and 100 GHz. We apply this
Bayesian scheme to white noise simulations and to more realistic simulations
that include CMB intensity, Galactic foregrounds and instrumental noise with
the characteristics of the QUIJOTE experiment Wide Survey at 11 GHz. Using
these simulations, we also compare our Bayesian method with the frequentist
Filtered Fusion method that has been already used in WMAP data and in the
\emph{Planck} mission. We find that the Bayesian method allows us to decrease
the threshold for a feasible estimation of $P$ to levels below $\sim 100$ mJy
(as compared to $\sim 500$ mJy that was the equivalent threshold for the
frequentist Filtered Fusion). We compare the bias introduced by the Bayesian
method and find it to be small in absolute terms. Finally, we test the
robustness of the Bayesian estimator against uncertainties in the prior and in
the flux density of the sources. We find that the Bayesian estimator is robust
against moderate changes in the parameters of the prior and almost insensitive
to realistic errors in the estimated photometry of the sources.
