  Likelihood-free methods are useful for parameter estimation of complex models
with intractable likelihood functions for which it is easy to simulate data.
Such models are prevalent in many disciplines including genetics, biology,
ecology and cosmology. Likelihood-free methods avoid explicit likelihood
evaluation by finding parameter values of the model that generate data close to
the observed data. The general consensus has been that it is most efficient to
compare datasets on the basis of a low dimensional informative summary
statistic, incurring information loss in favour of reduced dimensionality. More
recently, researchers have explored various approaches for efficiently
comparing empirical distributions in the likelihood-free context in an effort
to avoid data summarisation. This article provides a review of these full data
distance based approaches, and conducts the first comprehensive comparison of
such methods, both qualitatively and empirically. We also conduct a substantive
empirical comparison with summary statistic based likelihood-free methods. The
discussion and results offer guidance to practitioners considering a
likelihood-free approach. Whilst we find the best approach to be problem
dependent, we also find that the full data distance based approaches are
promising and warrant further development. We discuss some opportunities for
future research in this space. Computer code to implement the methods discussed
in this paper can be found at https://github.com/cdrovandi/ABC-dist-compare.
