Theoretical guarantees for informed MH when M < R < M^2

Determine whether informed Metropolis–Hastings algorithms on finite discrete spaces—specifically the informed proposals K_h constructed via weighting functions h(u) and acceptance probabilities as in Equation (19)—admit rigorous convergence guarantees when the unimodality/tail parameter R(X, N, π) satisfies M(X, N) < R(X, N, π) < M(X, N)^2. Clarify whether such informed samplers provably converge under this intermediate regime, which is not covered by the dimension-free relaxation time results requiring R(X, N, π) > M(X, N)^2.

Background

In the paper, X denotes a finite discrete state space with a neighborhood relation N and a target distribution π. The parameter M(X, N) is the maximum neighborhood size and R(X, N, π) captures unimodality and tail decay: if R > 1, π is unimodal with respect to N, and larger R implies faster tail decay.

The authors establish that random walk Metropolis–Hastings enjoys rapid mixing under the condition R > M (Theorem 3.1). For informed Metropolis–Hastings with bounded proposal weights (h(u) = clip(u, ℓ, L)), they prove dimension-free relaxation times under the stronger condition R > M^2 (Theorem 4.2), leveraging multicommodity flow and drift techniques.

The discussion raises an explicit open question about the intermediate regime M < R < M^2. While random walk MH mixes rapidly there, the paper’s current results for informed MH do not apply, leaving unclear whether any formal convergence guarantees can be established for informed samplers under this regime.