  It is expected that gravitational waves, similar to electromagnetic waves,
can be gravitationally lensed by intervening matters, producing multiple
instances of the same signal arriving at different times from different
apparent luminosity distances with different phase shifts compared to the
un-lensed signal due to lensing. If unaccounted for, these lensed signals will
masquerade as separate systems with higher mass and lower redshift. Here we
present a Bayesian statistical framework for identifying strongly-lensed
gravitational-wave signals that incorporates astrophysical information and
accounts for selection effects. We also propose a two-step hierarchical
analysis for more efficient computations of the probabilities and inferences of
source parameters free from bias introduced by lensing. We show with examples
on how changing the astrophysical models could shift one's interpretation on
the origin of the observed gravitational waves, and possibly lead to
indisputable evidence of strong lensing of the observed waves. In addition, we
demonstrate the improvement in the sky localization of the source of the lensed
signals, and in some cases the identification of the Morse indices of the
lensed signals. If confirmed, lensed gravitational waves will allow us to probe
the Universe at higher redshift, and to constrain the polarization contents of
the waves with fewer detectors.
