Abstract Deriving Bayesian inference for exponential random graph models (ERGMs) is a challenging “doubly intractable” problem as the normalizing constants of the likelihood and posterior density are both intractable. Markov chain Monte Carlo (MCMC) methods which yield Bayesian inference for ERGMs, such as the exchange algorithm, are asymptotically exact but computationally intensive, as a network has to be drawn from the likelihood at every step using, for instance, a “tie no tie” sampler. In this article, we develop a variety of variational methods for Gaussian approximation of the posterior density and model selection. These include nonconjugate variational message passing based on an adjusted pseudolikelihood and stochastic variational inference. To overcome the computational hurdle of drawing a network from the likelihood at each iteration, we propose stochastic gradient ascent with biased but consistent gradient estimates computed using adaptive self-normalized importance sampling. These methods provide attractive fast alternatives to MCMC for posterior approximation. We illustrate the variational methods using real networks and compare their accuracy with results obtained via MCMC and Laplace approximation. Supplementary materials for this article are available online.

中文翻译：

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指数随机图模型的贝叶斯变分推理

摘要 为指数随机图模型 (ERGM) 推导贝叶斯推理是一个具有挑战性的“双重棘手”问题，因为似然和后验密度的归一化常数都是棘手的。为 ERGM 产生贝叶斯推理的马尔可夫链蒙特卡罗 (MCMC) 方法，例如交换算法，渐近精确但计算量大，因为网络必须从每一步的似然性中提取，例如使用“tie no领带”采样器。在本文中，我们为后验密度和模型选择的高斯近似开发了各种变分方法。这些包括基于调整的伪似然和随机变分推理的非共轭变分消息传递。为了克服在每次迭代时根据似然绘制网络的计算障碍，我们提出了使用自适应自归一化重要性采样计算的有偏但一致的梯度估计的随机梯度上升。这些方法为 MCMC 的后验近似提供了有吸引力的快速替代方案。我们使用真实网络来说明变分方法，并将它们的准确性与通过 MCMC 和拉普拉斯近似获得的结果进行比较。本文的补充材料可在线获取。