  We perform a Bayesian analysis of neutrons star moment of inertia by
utilizing the available gravitational-wave data from LIGO/Virgo (GW170817 and
GW190425) and mass-radius measurements from the Neutron Star Interior
Composition Explorer (PSR J0030+0415 and PSR J0740+6620), incorporating the
possible phase transition in the pulsar inner core. We find that the moment of
inertia of pulsar A in the double pulsar binary J0737-3039 is
$\sim1.30\times10^{45}\,{\rm g\,cm^2}$, which only slightly depends on the
employed hadronic equation of states. We also demonstrate how a moment of
inertia measurement would improve our knowledge of the equation of state and
the mass-radius relation for neutron stars and discuss whether a quark
deconfinement phase transition is supported by the available data and
forthcoming data that could be consistent with this hypothesis. We find that if
pulsar A is a quark star, that its moment of inertia is a large value of
$\sim1.55\times10^{45}\,{\rm g\,cm^2}$ suggesting the possibility of
distinguishing it from (hybrid-)neutron stars with measurements of PSR
J0737-3039A moment of inertia. We finally demonstrate the
moment-of-inertia-compactness universal relations and provide analytical fits
for both (hybrid-)neutron star and quark star results based on our analysis.
