TipUpdate
The correct formulas can be found in [1], Section 2.3.
This is about an error on page 596, in the rejoinder of [2].
The error was pointed out by Anthony Lee. In short, the rejoinder contains a formula:
\[ \frac{1}{m-k+1}\sum_{t=k}^m h(X_t) + \sum_{l = k+L}^{\tau - 1}\min\left\{1, \frac{\lceil (l-k)/L\rceil}{m-k+1}\right\}\{h(X_{l})-h(Y_{l-L})\},\]
which is wrong, i.e. it does not do what it’s meant to do. It should read
\[ \frac{1}{m-k+1}\sum_{t=k}^m h(X_t) + \sum_{l = k+L}^{\tau - 1} \frac{\lfloor(l-k) / L\rfloor - \lceil \max(L, l-m)/L\rceil + 1}{m-k+1} \{h(X_{l})-h(Y_{l-L})\}.\]
In the notebook below, you can find more explanations and illustrative experiments.
https://github.com/pierrejacob/blog-code/blob/main/biascancellation.ipynb
[1]
Atchadé, Y. F. and Jacob, P. E. (2025+). Unbiased Markov Chain Monte Carlo: what, why, and how. To appear in the Handbook of Markov chain Monte Carlo (2nd edition).
[2]
Jacob, P. E., O’Leary, J. and Atchadé, Y. F. (2020). Unbiased markov chain monte carlo methods with couplings. Journal of the Royal Statistical Society Series B: Statistical Methodology 82 543–600.