Extending UNAM to countably infinite supports by computing f_m without full enumeration

Develop an algorithm to compute f_m = P^*(σ(m) → σ(m)) in the Upward Nested Antithetic Modification (UNAM) scheme without enumerating all m possible values, so that UNAM can be applied to discrete variables with countably infinite or very large support. Specifically, determine how to obtain the limiting form of the UNAM recursions and compute the self-transition probability for the most probable value σ(m) efficiently, thereby enabling practical sampling from the UNAM transition distribution in the infinite-support setting.

Background

UNAM is a nested antithetic modification method that orders focal values in non-decreasing probability and constructs modified transition probabilities that Peskun-dominate Gibbs sampling. While UNAM is straightforward for finite state spaces, the paper discusses the challenge of extending UNAM to variables with countably infinite or very large numbers of possible values.

The authors outline a potential approach—reversing recursions for quantities s_i and f_i to find a limiting form as m → ∞ and then sampling by computing only finitely many f_i as needed. However, they note a key obstacle: computing the self-transition probability for the most probable value, f_m = P^*(σ(m) → σ(m)), appears to require looking at all m values, which is infeasible in the infinite case.