  Generative adversarial networks (GANs) provide an algorithmic framework for
constructing generative models with several appealing properties: they do not
require a likelihood function to be specified, only a generating procedure;
they provide samples that are sharp and compelling; and they allow us to
harness our knowledge of building highly accurate neural network classifiers.
Here, we develop our understanding of GANs with the aim of forming a rich view
of this growing area of machine learning---to build connections to the diverse
set of statistical thinking on this topic, of which much can be gained by a
mutual exchange of ideas. We frame GANs within the wider landscape of
algorithms for learning in implicit generative models--models that only specify
a stochastic procedure with which to generate data--and relate these ideas to
modelling problems in related fields, such as econometrics and approximate
Bayesian computation. We develop likelihood-free inference methods and
highlight hypothesis testing as a principle for learning in implicit generative
models, using which we are able to derive the objective function used by GANs,
and many other related objectives. The testing viewpoint directs our focus to
the general problem of density ratio estimation. There are four approaches for
density ratio estimation, one of which is a solution using classifiers to
distinguish real from generated data. Other approaches such as divergence
minimisation and moment matching have also been explored in the GAN literature,
and we synthesise these views to form an understanding in terms of the
relationships between them and the wider literature, highlighting avenues for
future exploration and cross-pollination.
