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.
翻译:对指数随机图形模型(ERGMs)的推论是具有挑战性的“难以解决的”问题,因为可能性和后表密度的正常常数是难以解决的。Markov 链条Monte Carlo(MCMC)方法,这些方法产生ERGM(例如交换算法)的巴伊瑟式推论,这些方法在交换算法中是微不足道的,但在计算上是密集的,因为一个网络必须从每一个步骤的可能性中抽取,例如使用一个“无领带”取样器。在本篇文章中,我们为远表密度和模型选择的高斯近似近似提供了各种变异方法。这些方法包括非同质变异信息,通过调整的假象和随机变异推法传递。为了克服从每次转动的可能性中提取网络的计算障碍,我们建议采用偏差梯度梯度梯度梯度,并使用适应性自我调整的重要性取样计算出偏差但一致的梯度估计值。这些方法为MCMC提供具有吸引力的近似性快速替代方法,用于后表象准。我们用实际网络和近似率比较的方法说明变异性方法。我们用实际的模型比较了变式方法,并比较了其结果。