An Insight Into the Gibbs Sampler: Keep the Samples or Drop Them?

In this letter, we propose an insight into Markov Chain Monte Carlo (MCMC) algorithms and more precisely the Gibbs sampler. From a didactic toy model, based on a normal bivariate distribution, a non-asymptotic analysis is derived and estimators are fully characterized. It provides a worthwhile and non-empirical understanding of the Gibbs sampler behaviour. Issues are investigated, such as the influence of the "burn-in" phase, useful in practice. Especially, the trade-off between discarding samples and integrating them into estimators is studied. On the whole, it leads to an analytical awareness of MCMC sampler.