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：不可计算归一化函数模型具有广泛的应用。由于归一化常数是关注参数的函数，标准马尔可夫链蒙特卡罗方法无法用于这类模型的贝叶斯推断。目前已有多种算法被开发用于处理此类模型。其中一些算法以后验分布作为其渐近分布，而其他"渐近非精确"算法则不具备这一性质。目前缺乏评估基于这些算法获得的近似结果的指导方法，因此很难对这些算法进行调优。我们针对不可计算归一化函数模型提出了两种新的诊断方法。第一种诊断方法受第二巴特利特恒等式的启发，原则上可广泛适用于超越归一化函数问题的蒙特卡洛近似。我们开发了该诊断方法的近似版本，使其适用于不可计算归一化函数问题。第二种诊断方法是基于Gorham和Mackey（2017）提出的核斯坦因不一致性诊断的蒙特卡洛近似。我们为这些方法提供了理论依据，并将其应用于多个具有挑战性的模拟和真实数据示例中的算法，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，在这些情境下获得了关于算法的有趣见解。