Convergence of Independence Metropolis-Hastings Algorithms in High Dimensions: Iteration Lengths, Cutoff Profiles, and Diagnostic Unreliability
Abstract
We characterize the high-dimensional convergence of the arithmetic mean from the independence Metropolis-Hastings algorithm initialized at its proposal. Convergence is measured in the worst-case expected absolute error over bounded functions. We prove necessary and sufficient conditions for the mean to converge in relation to the importance weight. We precisely describe the logarithmic cutoff profile of the sharp decay to convergence where the relative entropy between the target and proposal determines the required iteration length. These results make connections between the required iterations of independence Metropolis-Hastings and importance sampling. For concentrating Bayesian posteriors, we illustrate how the required iteration time can be estimated providing concrete guidance for simulations. We justify the logarithmic cutoff analysis by showing a standard cutoff phenomenon is absent and prove that popular convergence diagnostics based on simulation output can validate inaccurate simulations.
Related
- Causal Discovery beyond Conditional Independence: Identifiability, Algorithms, and Statistical Inference for Directed Acyclic and Cyclic Graphs
- Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence
- A note on the convergence guarantees of RLT-based algorithms for polynomial optimization
- Monte Carlo Power Iteration Algorithms: Convergence, Analysis and Application to Reactor Criticality Calculations
- Volume of quasi-homogeneous sublevel sets: Two linear algebra deterministic algorithms with convergence rates
BibTeX
@misc{brown2026convergence,
title = {{Convergence of Independence Metropolis-Hastings Algorithms in High Dimensions: Iteration Lengths, Cutoff Profiles, and Diagnostic Unreliability}},
author = {Austin Brown},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34531},
eprint = {2610.34531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34531},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convergence of Independence Metropolis-Hastings Algorithms in High Dimensions: Iteration Lengths, Cutoff Profiles, and Diagnostic Unreliability
AU - Austin Brown
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34531
UR - https://arxiv.org/abs/2610.34531
AB - We characterize the high-dimensional convergence of the arithmetic mean from the independence Metropolis-Hastings algorithm initialized at its proposal. Convergence is measured in the worst-case expected absolute error over bounded functions. We prove necessary and sufficient conditions for the mean to converge in relation to the importance weight. We precisely describe the logarithmic cutoff profile of the sharp decay to convergence where the relative entropy between the target and proposal determines the required iteration length. These results make connections between the required iterations of independence Metropolis-Hastings and importance sampling. For concentrating Bayesian posteriors, we illustrate how the required iteration time can be estimated providing concrete guidance for simulations. We justify the logarithmic cutoff analysis by showing a standard cutoff phenomenon is absent and prove that popular convergence diagnostics based on simulation output can validate inaccurate simulations.
ER -