Exponential random graph models (ERGMs) are a widely used framework for network data, enabling hypothesis testing on the structural mechanisms underlying observed networks. Bayesian ERGMs provide principled uncertainty quantification and enable the incorporation of prior knowledge through fully probabilistic modelling. However, computation remains challenging because the posterior is doubly intractable, with a likelihood normalising constant that depends on unknown parameters. This paper reviews Bayesian approaches to ERGM inference, categorising inference methods into three broad classes: auxiliary variable MCMC methods, adjusted pseudo-likelihood approaches, and variational methods, alongside dedicated treatment of model selection. We also discuss modelling extensions for missing data, longitudinal dynamics, populations of networks, weighted networks, highlighting applications across various scientific disciplines.

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