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 and density-difference estimation. There are four approaches for density comparison, 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, 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. * Equal contribution 1 DeepMind, London, UK. Correspondence to: Shakir Mohamed <shakir@google.com>, Balaji Lakshminarayanan <balajiln@google.com>. It is useful to make a distinction between two types of probabilistic models: prescribed and implicit models (Diggle and Gratton, 1984) . Prescribed probabilistic models are those that provide an explicit parametric specification of the distribution of an observed random variable x, specifying a log-likelihood function log q θ (x) with parameters θ. Most models in machine learning and statistics are of this form, whether they be state-of-the-art classifiers for object recognition, complex sequence models for machine translation, or fine-grained spatio-temporal models tracking the spread of disease. Alternatively, we can specify implicit probabilistic models that define a stochastic procedure that directly generates data. Such models are the natural approach for problems in climate and weather, population genetics, and ecology, since the mechanistic understanding of such systems can be used to directly create a data simulator, and hence the model. It is exactly because implicit models are more natural for many problems that they are of interest and importance. Implicit generative models use a latent variable z and transform it using a deterministic function G θ that maps from R m → R d using parameters θ. Such models are amongst the most fundamental of models, e.g., many of the basic methods for generating non-uniform random variates are based on simple implicit models and one-line transformations (Devroye, 2006) . In general, implicit generative models specify a valid density on the output space that forms an effective likelihood function: where q(z) is a latent variable that provides the external source of randomness and equation (2) is the definition of the transformed density as the derivative of the cumulative distribution function. When the function G is well-defined, such as when the function is invertible, or has dimensions m = d with easily characterised roots, we recover the familiar rule for transformations of probability distributions. We are interested in developing more general and flexible implicit generative models where the function G is a nonlinear function with d > m, specified by deep networks.
Borrowing terminology from REF , GANs can be seen as implicit probabilistic models that are directly optimizing for generating visually appealing samples as opposed to prescribed probabilistic models such as BGMs that provide an explicit characterization of the log-likelihood and are typically stable to train.
Learning in Implicit Generative Models
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