Efficient Computational Strategies for Doubly Intractable Problems with Applications to Bayesian Social Networks

Powerful ideas recently appeared in the literature are adjusted and combined to design improved samplers for doubly intractable target distributions with a focus on Bayesian exponential random graph models. Different forms of adaptive Metropolis–Hastings proposals (vertical, horizontal and rectangular) are tested and merged with the delayed rejection (DR) strategy with the aim of reducing the variance of the resulting Markov chain Monte Carlo estimators for a given computational time. The DR is modified in order to integrate it within the approximate exchange algorithm (AEA) to avoid the computation of intractable normalising constant that appears in exponential random graph models. This gives rise to the AEA + DR: a new methodology to sample doubly intractable distributions that dominates the AEA in the Peskun ordering (Peskun Biometrika 60:607– 612, 1973) leading to MCMC estimators with a smaller asymptotic variance. The Bergm package for R (Caimo and Friel J. Stat. Softw. 22:518–532, 2014) has been updated to incorporate the AEA + DR thus including the possibility of adding a higher stage proposals and different forms of adaptation.