Bayesian Inference in the Presence of Intractable Normalizing Functions

Models with intractable normalizing functions arise frequently in statistics. Common examples of such models include exponential random graph models for social networks and Markov point processes for ecology and disease modeling. Inference for these models is complicated because the normalizing functions of their probability distributions include the parameters of interest. In Bayesian analysis, they result in so-called doubly intractable posterior distributions which pose significant computational challenges. Several Monte Carlo methods have emerged in recent years to address Bayesian inference for such models. We provide a framework for understanding the algorithms, and elucidate connections among them. Through multiple simulated and real data examples, we compare and contrast the computational and statistical efficiency of these algorithms and discuss their theoretical bases. Our study provides practical recommendations for practitioners along with directions for future research for Markov chain Monte Carlo (MCMC) methodologists. Supplementary materials for this article are available online.

具有难以处理的归一化函数的模型在统计学领域中频繁出现。这类模型的常见示例包括面向社交网络的指数随机图模型（Exponential Random Graph Models），以及应用于生态学与疾病建模的马尔可夫点过程（Markov Point Processes）。此类模型的统计推断工作较为复杂，原因在于其概率分布的归一化函数中包含了研究者关注的参数。在贝叶斯分析框架下，这类模型会产生所谓的双重难以处理的后验分布，带来了严峻的计算挑战。近年来，已有多种蒙特卡洛（Monte Carlo）方法被提出，用于解决此类模型的贝叶斯推断问题。本文构建了一套用于理解这些算法的统一分析框架，并阐明了各算法之间的内在关联。本文通过多组模拟数据与真实数据案例，对比分析了这些算法的计算效率与统计效率，并探讨了其理论基础。本研究不仅为实际应用从业者提供了实用的操作建议，也为马尔可夫链蒙特卡洛（Markov Chain Monte Carlo，MCMC）方法研究者指明了未来的研究方向。本文的补充材料可在线获取。