  Gravitational-wave (GW) measurements of physical effects such as spin-induced
quadrupole moments can distinguish binaries consisting of black holes from
non-black hole binaries. While these effects may be poorly constrained for
single-event inferences with the second-generation detectors, combining
information from multiple detections can help uncover features of non-black
hole binaries. The spin-induced quadrupole moment has specific predictions for
different types of compact objects, and a generalized formalism must consider a
population where different types of compact objects co-exist. In this study, we
introduce a hierarchical mixture-likelihood formalism to estimate the {\it
fraction of non-binary black holes in the population}. We demonstrate the
applicability of this method using simulated GW signals injected into Gaussian
noise following the design sensitivities of the Advanced LIGO Advanced Virgo
detectors. We compare the performance of this method with a
traditionally-followed hierarchical inference approach. Both the methods are
equally effective to hint at inhomogeneous populations, however, we find the
mixture-likelihood approach to be more natural for mixture populations
comprising compact objects of diverse classes. We also discuss the possible
systematics in the mixture-likelihood approach, caused by several reasons,
including the limited sensitivity of the second-generation detectors, specific
features of the astrophysical population distributions, and the limitations
posed by the waveform models employed. Finally, we apply this method to the
LIGO-Virgo detections published in the second GW transient catalog (GWTC-2) and
find them consistent with a binary black hole population within the statistical
precision.
