A long-standing gap exists between the theoretical analysis of Markov chain Monte Carlo convergence, which is often based on statistical divergences, and the diagnostics used in practice. We introduce the first general convergence diagnostics for Markov chain Monte Carlo based on any $f$-divergence, allowing users to directly monitor, among others, the Kullback-Leibler and the $χ^2$ divergences as well as the Hellinger and the total variation distances. Our approach rests on a coupling-based "weight harmonization" scheme that produces direct, computable, and consistent importance weights for interacting Markov chains with respect to their target distribution. Beyond their use as convergence diagnostics, these weights are consistent estimates of the Radon-Nikodym derivative $\mathrm{d}π/\mathrm{d} μ_t$, a richer object than the convergence bounds alone, with natural applications to importance-weighted inference. We show how such weightings can provide upper bounds to any $f$-divergence, prove that these bounds tighten over time and converge to zero as the chains approach stationarity, and demonstrate that, while more conservative than existing coupling-based total variation estimators, our method remains a practical and broadly applicable diagnostic tool.

翻译：马尔可夫链蒙特卡罗方法的理论收敛性分析（通常基于统计散度）与实际使用的诊断工具之间长期存在差距。我们首次提出一种基于任意$f$-散度的马尔可夫链蒙特卡罗通用收敛诊断方法，使用户能够直接监测包括KL散度、$\chi^2$散度、Hellinger距离和总变差距离在内的多种指标。我们的方法基于一种耦合型“权重协调”机制，该机制能为相互作用的马尔可夫链产生关于其目标分布的直接、可计算且一致的的重要性权重。除了作为收敛诊断工具外，这些权重还是Radon-Nikodym导数$\mathrm{d}π/\mathrm{d} μ_t$的一致估计——这一对象比单纯的收敛界更为丰富，可自然应用于重要性加权推断。我们展示了此类加权如何能为任意$f$-散度提供上界，证明这些上界随时间收紧并在链接近平稳时收敛至零，并论证尽管我们的方法比现有基于耦合的总变差估计量更为保守，但它仍然是一种实用且广泛适用的诊断工具。