  The discovery of GW170817 provided the first direct gravitational-wave
measurement of the Hubble constant, $H_0$, demonstrating the potential power of
standard-siren cosmology. The dark siren approach can be utilized for
gravitational-wave sources in the absence of an electromagnetic counterpart:
one considers all galaxies contained within the localization volume as
potential hosts. When statistically averaging over the potential host galaxies,
one can weight them by their luminosities to account for physically-motivated
prescriptions (e.g., tracing star formation or stellar mass). Using mock galaxy
catalogs, we explore the impact of these weightings on the measurement of
$H_0$, focusing on the bias in $H_0$ inference that results from
incorrectly-weighted prescriptions. We find that assuming an incorrect galaxy
host probability can lead to significant biases in $H_0$, up to about five
times off from typical values inferred by current experiments. These biases are
due to inconsistent galaxy weighted redshift distributions as well as
preferentially selecting the incorrect host during inference. The magnitudes of
these biases are influenced by the galaxy number density along each line of
sight, the uncertainty in the measurement of the gravitational-wave luminosity
distance, and correlations in the parameter space of galaxies. These biases may
be overcome with future GW detectors that contain better GW localization, using
a strategic choice of weighting prescription, or with increasing the SNR cut.
We propose using hierarchical inference as a diagnosis of incorrectly-weighted
prescriptions, which can further be used to simultaneously infer the correct
weighting scheme and cosmological parameters.
