# Nested hat R: Assessing the convergence of Markov chain Monte Carlo when running many short chains

arXiv: https://arxiv.org/abs/2110.13017 · HTML: https://mlsanity.com/paper/2110.13017/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2021.

## Built on by (most cited first, 9 of 9)

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- [Disentangling impact of capacity, objective, batchsize, estimators, and step-size on flow VI](https://mlsanity.com/paper/2412.08824/) (2024)
- [Amortized Bayesian Workflow](https://mlsanity.com/paper/2409.04332/) (2024)
- [Comparison of parallel SMC and MCMC for Bayesian deep learning](https://mlsanity.com/paper/2402.06173/) (2024)
- [Scalable Bayesian Monte Carlo: fast uncertainty estimation beyond deep ensembles](https://mlsanity.com/paper/2505.13585/) (2025)
- [Asymptotically exact variational flows via involutive MCMC kernels](https://mlsanity.com/paper/2506.02162/) (2025)

## Abstract (arXiv)

Recent developments in parallel Markov chain Monte Carlo (MCMC) algorithms allow us to run thousands of chains almost as quickly as a single chain, using hardware accelerators such as GPUs. While each chain still needs to forget its initial point during a warmup phase, the subsequent sampling phase can be shorter than in classical settings, where we run only a few chains. To determine if the resulting short chains are reliable, we need to assess how close the Markov chains are to their stationary distribution after warmup. The potential scale reduction factor widehat R is a popular convergence diagnostic but unfortunately can require a long sampling phase to work well. We present a nested design to overcome this challenge and a generalization called nested widehat R. This new diagnostic works under conditions similar to widehat R and completes the workflow for GPU-friendly samplers. In addition, the proposed nesting provides theoretical insights into the utility of widehat R, in both classical and short-chains regimes.