  We propose a Bayesian inference framework to predict the merger history of
LIGO-Virgo binary black holes, whose binary components may have undergone
hierarchical mergers in the past. The framework relies on numerical relativity
predictions for the mass, spin, and kick velocity of the remnant black holes.
This proposed framework computes the masses, spins, and kicks imparted to the
remnant of the parent binaries, given the initial masses and spin magnitudes of
the binary constituents. We validate our approach by performing an "injection
study" based on a constructed sequence of hierarchically-formed binaries. Noise
is added to the final binary in the sequence, and the parameters of the
'parent' and 'grandparent' binaries in the merger chain are then reconstructed.
This method is then applied to three GWTC-3 events: GW190521, GW200220_061928,
GW190426_190642. These events were selected because at least one of the binary
companions lies in the putative pair-instability supernova mass gap, in which
stellar processes alone cannot produce black holes. Hierarchical mergers offer
a natural explanation for the formation of black holes in the pair-instability
mass-gap. We use the backward evolution framework to predict the parameters of
the parents of the primary companion of these three binaries. Our results
indicate that at least one component of these three observed binaries was
formed through a prior binary black hole merger. This approach can be readily
applied to future high-mass gravitational wave events to predict their
formation history under the hierarchical merger assumption.
