  Bayesian inference for models that have an intractable partition function is
known as a doubly intractable problem, where standard Monte Carlo methods are
not applicable. The past decade has seen the development of auxiliary variable
Monte Carlo techniques (M{\o}ller et al., 2006; Murray et al., 2006) for
tackling this problem; these approaches being members of the more general class
of pseudo-marginal, or exact-approximate, Monte Carlo algorithms (Andrieu and
Roberts, 2009), which make use of unbiased estimates of intractable posteriors.
Everitt et al. (2017) investigated the use of exact-approximate importance
sampling (IS) and sequential Monte Carlo (SMC) in doubly intractable problems,
but focussed only on SMC algorithms that used data-point tempering. This paper
describes SMC samplers that may use alternative sequences of distributions, and
describes ways in which likelihood estimates may be improved adaptively as the
algorithm progresses, building on ideas from Moores et al. (2015). This
approach is compared with a number of alternative algorithms for doubly
intractable problems, including approximate Bayesian computation (ABC), which
we show is closely related to the method of M{\o}ller et al. (2006).
