MCMC Can See Can't See
Abstract
It is folklore that passing "convergence" diagnostics does not guarantee that a Markov chain has converged. We prove this as an impossibility for any finite checklist of diagnostics. Running more chains, longer chains, and smarter diagnostics does not avoid this impossibility in certifying convergence. Under regularity assumptions, we determine how the worst-case undetected non-convergence decays as more diagnostics are added, and how to choose diagnostics optimally. Folklore suggests that the sampler's slowest mixing modes are the "dangerous" ones that must be monitored; we prove that the slow modes yield nearly optimal diagnostics as sampling proceeds. Given a source class, we also give an anytime-valid bound on the remaining discrepancy and show how undetected bias caps the effective sample size that a finite checklist can certify.
BibTeX
@misc{smola2026mcmc,
title = {{MCMC Can See Can't See}},
author = {Daniel Matsui Smola},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20020},
eprint = {2610.20020},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20020},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - MCMC Can See Can't See
AU - Daniel Matsui Smola
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20020
UR - https://arxiv.org/abs/2610.20020
AB - It is folklore that passing "convergence" diagnostics does not guarantee that a Markov chain has converged. We prove this as an impossibility for any finite checklist of diagnostics. Running more chains, longer chains, and smarter diagnostics does not avoid this impossibility in certifying convergence. Under regularity assumptions, we determine how the worst-case undetected non-convergence decays as more diagnostics are added, and how to choose diagnostics optimally. Folklore suggests that the sampler's slowest mixing modes are the "dangerous" ones that must be monitored; we prove that the slow modes yield nearly optimal diagnostics as sampling proceeds. Given a source class, we also give an anytime-valid bound on the remaining discrepancy and show how undetected bias caps the effective sample size that a finite checklist can certify.
ER -