Rigorous confidence bounds for MCMC under a geometric drift condition

interest and Ît,n = (1/n) ∑t+n−1 i=t f(Xi) its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory n and burn-in time t which ensure that P (|Ît,n − I| ≤ ε) ≥ 1− α. The bounds depend only and explicitly on drift parameters, on the V−norm of f, where V is the drift function and on precision and confidence parameters ε, α. Next we analyse an MCMC estimator based on the median of multiple shorter runs that allows for sharper bounds for the required total simulation cost. In particular the methodology can be applied for computing Bayesian estimators in practically relevant models. We illustrate our bounds numerically in a simple example.