  Powerful ideas recently appeared in the literature are adjusted and combined
to design improved samplers for Bayesian exponential random graph models.
Different forms of adaptive Metropolis-Hastings proposals (vertical, horizontal
and rectangular) are tested and combined 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. In the examples treated
in this paper the best combination, namely horizontal adaptation with delayed
rejection, leads to a variance reduction that varies between 92% and 144%
relative to the adaptive direction sampling approximate exchange algorithm of
Caimo and Friel (2011). These results correspond to an increased performance
which varies from 10% to 94% if we take simulation time into account. The
highest improvements are obtained when highly correlated posterior
distributions are considered.
