  Doubly intractable models are encountered in a number of fields, e.g. social
networks, ecology and epidemiology. Inference for such models requires the
evaluation of a likelihood function, whose normalising function depends on the
model parameters and is typically computationally intractable. The normalising
constant of the posterior distribution and the additional normalising function
of the likelihood function result in a so-called doubly intractable posterior,
for which it is difficult to directly apply Markov chain Monte Carlo (MCMC)
methods. We propose a signed pseudo-marginal Metropolis-Hastings (PMMH)
algorithm with an unbiased block-Poisson estimator to sample from the posterior
distribution of doubly intractable models. As the estimator can be negative,
the algorithm targets the absolute value of the estimated posterior and uses an
importance sampling correction to ensure simulation consistent estimates of the
posterior mean of any function. The advantages of our estimator over previous
approaches are that its form is ideal for correlated pseudo-marginal methods
which are well known to dramatically increase sampling efficiency. Moreover, we
develop analytically derived heuristic guidelines for optimally tuning the
hyperparameters of the estimator. We demonstrate the algorithm on the Ising
model and a Kent distribution model for spherical data.
