In this paper we discuss a well known computing problem -- inference for models with intractable normalizing functions. Models with intractable normalizing functions arise in a wide variety of areas, for instance network models, models for spatial data on lattices, spatial point processes, flexible models for count data and gene expression, and models for permutations. Simulating from these models for fixed parameter values is well studied, starting with work dating back seventy years to the origin of the Metropolis algorithm. On the other hand some of the most practical and theoretically justified algorithms for inference, particularly Bayesian inference, have only been developed within the past two decades. The most computationally efficient algorithms often do not have well developed theory and few if any approaches exist for assessing the quality of approximations based on them. For many problems even the best algorithms can be computationally infeasible. Hence, this is an exciting area of research with many open problems. We explain several key algorithms, providing connections and touching upon practical advantages and disadvantages of each, with some discussion of theoretical properties where they impact practice. We discuss an approach for assessing the accuracy of approximations produced by these algorithms; this diagnostic is particularly valuable for algorithm tuning. While our focus is largely on models with intractable normalizing functions, we also discuss algorithms that are more broadly applicable to models where the entire likelihood function is intractable; these methods are of course also applicable to intractable normalizing function problems.

翻译：本文探讨一个广为人知的计算问题——具有难处理归一化函数模型的推断。此类模型广泛出现于诸多领域，例如网络模型、格点空间数据模型、空间点过程、计数数据与基因表达的灵活模型，以及排列模型。从这些模型中针对固定参数值进行模拟的研究已相当深入，其起源可追溯至七十年前Metropolis算法的提出。然而，一些最实用且理论依据充分的推断算法（尤其是贝叶斯推断）仅在近二十年内才得以发展。计算效率最高的算法往往缺乏完善的理论支撑，且用于评估基于这些算法的近似解质量的现有方法极少甚至不存在。对于许多问题，即便最优算法在计算上也可能不可行。因此，这是一个充满开放性问题、令人振奋的研究领域。本文阐释了若干关键算法，揭示其内在联系并评述各自的实践优缺点，同时就影响实践的理论特性展开讨论。我们探讨了一种评估这些算法所生成近似解准确性的方法；该诊断工具对于算法调优尤为宝贵。尽管研究重点主要集中于具有难处理归一化函数的模型，我们也讨论了更广泛适用于整个似然函数难处理模型的算法；这些方法当然也适用于难处理归一化函数问题。