precision in MCMC

While browsing Images des Mathématiques, I came across this article [in French] that studies the impact of round-off errors on number representations in a dynamical system and checked how much this was the case for MCMC algorithms like the slice sampler (recycling some R code from Monte Carlo Statistical Methods). By simply adding a few signif(…,dig=n) in the original R code. And letting the precision n vary.

“…si on simule des trajectoires pendant des intervalles de temps très longs, trop longs par rapport à la précision numérique choisie, alors bien souvent, les résultats des simulations seront complètement différents de ce qui se passe en réalité…” Pierre-Antoine Guihéneuf

Rather unsurprisingly (!), using a small enough precision (like two digits on the first row) has a visible impact on the simulation of a truncated normal. Moving to three digits seems to be sufficient in this example… One thing this tiny experiment reminds me of is the lumpability property of Kemeny and Snell.  A restriction on Markov chains for aggregated (or discretised) versions to be ergodic or even Markov. Also, in 2000, Laird Breyer, Gareth Roberts and Jeff Rosenthal wrote a Statistics and Probability Letters paper on the impact of round-off errors on geometric ergodicity. However, I presume [maybe foolishly!] that the result stated in the original paper, namely that there exists an infinite number of precision digits for which the dynamical system degenerates into a small region of the space does not hold for MCMC. Maybe foolishly so because the above statement means that running a dynamical system for “too” long given the chosen precision kills the intended stationary properties of the system. Which I interpret as getting non-ergodic behaviour when exceeding the period of the uniform generator. More or less.