# MCMC Can See Can't See

arXiv: https://arxiv.org/abs/2610.20020 · HTML: https://mlsanity.com/paper/2610.20020/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 37 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Builds on (most cited first, 4 of 4)

- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence](https://mlsanity.com/paper/2004.00891/) (2020)
- [Hypothesis testing for Markov chain Monte Carlo](https://mlsanity.com/paper/1409.7986/) (2014)
- [Measuring and Decomposing Mode Separation via the Canonical Diffusion](https://mlsanity.com/paper/2605.08777/) (2026)

## Abstract (arXiv)

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.