Models with intractable normalizing functions have numerous applications. Because the normalizing constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. A number of algorithms have been developed for such models. Some have the posterior distribution as their asymptotic distribution. Other ``asymptotically inexact'' algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms. Hence it is very hard to tune them. We propose two new diagnostics that address these problems for intractable normalizing function models. Our first diagnostic, inspired by the second Bartlett identity, is in principle broadly applicable to Monte Carlo approximations beyond the normalizing function problem. We develop an approximate version of this diagnostic that is applicable to intractable normalizing function problems. Our second diagnostic is a Monte Carlo approximation to a kernel Stein discrepancy-based diagnostic introduced by Gorham and Mackey (2017). We provide theoretical justification for our methods and apply them to several algorithms in challenging simulated and real data examples including an Ising model, an exponential random graph model, and a Conway--Maxwell--Poisson regression model, obtaining interesting insights about the algorithms in these contexts.

翻译：具有难以计算的归一化函数的模型在许多应用中具有重要意义。由于归一化常数是感兴趣参数的函数，标准马尔可夫链蒙特卡罗方法无法用于这些模型的贝叶斯推断。针对此类模型，已开发出多种算法。其中一些算法以后验分布作为其渐近分布，而其他“渐近不精确”算法则不具备这一性质。目前关于基于这些算法的近似评估的指导十分有限，因此对其进行调优极为困难。我们提出了两种新的诊断方法，以解决难以计算的归一化函数模型中的这些问题。第一种诊断方法受第二巴特莱特恒等式启发，原则上可广泛适用于归一化函数问题之外的蒙特卡罗近似。我们开发了该诊断方法的近似版本，适用于难以计算的归一化函数问题。第二种诊断方法是对戈勒姆和麦基（2017）提出的基于核斯坦因差异的蒙特卡罗近似诊断。我们为所提出的方法提供了理论依据，并将其应用于多个算法中，涵盖具有挑战性的模拟与真实数据示例，包括伊辛模型、指数随机图模型以及康威-麦克斯韦-泊松回归模型，从而在这些情境下获得了关于算法性能的有趣见解。