  We address the problem of upper bounding the mean square error of MCMC
estimators. Our analysis is nonasymptotic. We first establish a general result
valid for essentially all ergodic Markov chains encountered in Bayesian
computation and a possibly unbounded target function $f$. The bound is sharp in
the sense that the leading term is exactly $\sigma_{\mathrm {as}}^2(P,f)/n$,
where $\sigma_{\mathrm{as}}^2(P,f)$ is the CLT asymptotic variance. Next, we
proceed to specific additional assumptions and give explicit computable bounds
for geometrically and polynomially ergodic Markov chains under quantitative
drift conditions. As a corollary, we provide results on confidence estimation.
