  Gravitational wave (GW) sources at cosmological distances can be used to
probe the expansion rate of the Universe. GWs directly provide a distance
estimation of the source but no direct information on its redshift. The optimal
scenario to obtain a redshift is through the direct identification of an
electromagnetic (EM) counterpart and its host galaxy. With almost 100 GW
sources detected without EM counterparts (dark sirens), it is becoming crucial
to have statistical techniques able to perform cosmological studies in the
absence of EM emission. Currently, only two techniques for dark sirens are used
on GW observations: the spectral siren method, which is based on the
source-frame mass distribution to estimate conjointly cosmology and the
source's merger rate, and the galaxy survey method, which uses galaxy surveys
to assign a probabilistic redshift to the source while fitting cosmology. It
has been recognized, however, that these two methods are two sides of the same
coin. In this paper, we present a novel approach to unify these two methods. We
apply this approach to several observed GW events using the \textsc{glade+}
galaxy catalog discussing limiting cases. We provide estimates of the Hubble
constant, modified gravity propagation effects, and population properties for
binary black holes. We also estimate the binary black hole merger rate per
galaxy to be $10^{-6}-10^{-5} {\rm yr^{-1}}$ depending on the galaxy catalog
hypotheses.
