Algorithm for computing the Lévy–Prokhorov distance needed for MH tuning

Develop an efficient algorithm to compute the Lévy–Prokhorov distance between probability measures on continuous subsets of R^d so that the tuning scheme that maximizes a large-deviation-based rate function lower bound over multiple test measures at prescribed distance from the target distribution π (Algorithm 3 in this paper) can be implemented in practice.

Background

The paper proposes large deviation-based tuning schemes for Metropolis–Hastings algorithms. One scheme (Algorithm 3) selects multiple test measures μ_i whose Lévy–Prokhorov distances from the target π are approximately a chosen ε, and then optimizes a rate-function lower bound over these measures. Implementing this scheme requires computing or approximating the Lévy–Prokhorov distance between probability measures.

The authors note that, despite the conceptual advantages of this scheme, a practical roadblock is the lack of an efficient method to compute the Lévy–Prokhorov metric in this setting, and they explicitly defer the development of such an algorithm to future research.