Neural posterior estimation on exponential random graph models: evaluating bias and implementation challenges

Exponential random graph models (ERGMs) are flexible probabilistic frameworks to model statistical networks through a variety of network summary statistics. Conventional Bayesian estimation for ERGMs involves iteratively exchanging with an auxiliary variable due to the intractability of the ERGM likelihood. However, this approach has limited scalability in large-scale implementations. Neural posterior estimation (NPE) is a recent advancement in simulation-based inference, using a neural network-based density estimator to infer the posterior for models with doubly intractable likelihoods for which simulations can be generated. While NPE has been successfully adopted in various fields such as cosmology, little research has investigated its use for ERGMs. Performing NPE on ERGMs not only provides a different approach to estimation for intractable ERGM likelihoods but also allows more efficient and scalable inference using the amortisation properties of NPE, and therefore we investigate how NPE can be effectively implemented in ERGMs. In this study, we present the first systematic implementation of NPE for ERGMs, rigorously evaluating potential biases, interpreting the bias magnitudes, and assessing computational costs. We compare NPE fits with conventional Bayesian ERGM fits as well as related neural simulation-based methods, namely neural likelihood estimation and neural ratio estimation. In our synthetic data analysis, we show that training a neural posterior estimator on 500,000 simulations circumvents the roughly 4,000,000,000 simulations required by conventional exchange-algorithm inference, enabling real-time posterior estimation. More importantly, our work highlights ERGM-specific areas that may pose particular challenges for the adoption of NPE.