  Extending prior work by Pankow et al, we introduce RIFT, an algorithm to
perform Rapid parameter Inference on gravitational wave sources via Iterative
Fitting. We demonstrate this approach can correctly recover the parameters of
coalescing compact binary systems, using detailed comparisons of RIFT to the
well-tested LALInference software library. We provide several examples where
the unique speed and flexibility of RIFT enables otherwise intractable or
awkward parameter inference analyses, including (a) adopting either costly and
novel models for outgoing gravitational waves; and (b) mixed approximations,
each suitable to different parts of the compact binary parameter space. We
demonstrate how \RIFT{} can be applied to binary neutron stars, both for
parameter inference and direct constraints on the nuclear equation of state.
