  A large number of statistical models are "doubly-intractable": the likelihood
normalising term, which is a function of the model parameters, is intractable,
as well as the marginal likelihood (model evidence). This means that standard
inference techniques to sample from the posterior, such as Markov chain Monte
Carlo (MCMC), cannot be used. Examples include, but are not confined to,
massive Gaussian Markov random fields, autologistic models and Exponential
random graph models. A number of approximate schemes based on MCMC techniques,
Approximate Bayesian computation (ABC) or analytic approximations to the
posterior have been suggested, and these are reviewed here. Exact MCMC schemes,
which can be applied to a subset of doubly-intractable distributions, have also
been developed and are described in this paper. As yet, no general method
exists which can be applied to all classes of models with doubly-intractable
posteriors. In addition, taking inspiration from the Physics literature, we
study an alternative method based on representing the intractable likelihood as
an infinite series. Unbiased estimates of the likelihood can then be obtained
by finite time stochastic truncation of the series via Russian Roulette
sampling, although the estimates are not necessarily positive. Results from the
Quantum Chromodynamics literature are exploited to allow the use of possibly
negative estimates in a pseudo-marginal MCMC scheme such that expectations with
respect to the posterior distribution are preserved. The methodology is
reviewed on well-known examples such as the parameters in Ising models, the
posterior for Fisher-Bingham distributions on the $d$-Sphere and a large-scale
Gaussian Markov Random Field model describing the Ozone Column data. This leads
to a critical assessment of the strengths and weaknesses of the methodology
with pointers to ongoing research.
