Mechanism behind near-exactness of approximate exchange algorithms with large internal error

Characterize and explain the phenomenon by which certain approximations to the exchange algorithm for doubly-intractable distributions yield stationary distributions that remain close to the target even when the internal approximation within the algorithm is far from convergence, and derive conditions under which this behavior occurs.

Background

Empirical results indicate that a particular approximation to the exchange algorithm can have a stationary distribution very close to the true target even when its internal approximation is far from converged, a regime where classical small-perturbation theory should not apply.

The authors highlight this as an explicit open question, seeking a theoretical explanation for the observed robustness in the large-perturbation regime.