  Using concentration inequalities, we give non-asymptotic confidence intervals
for estimates obtained by Markov chain Monte Carlo (MCMC) simulations, when
using the approximation $\mathbb{E}_{\pi} f\approx (1/(N-t_0))\cdot
\sum_{i=t_0+1}^N f(X_i)$. To allow the application of non-asymptotic error
bounds in practice, here we state bounds formulated in terms of the spectral
properties of the chain and the properties of $f$ and propose estimators of the
parameters appearing in the bounds, including the spectral gap, mixing time,
and asymptotic variance. We introduce a method for setting the burn-in time and
the initial distribution that is theoretically well-founded and yet is
relatively simple to apply. We also investigate the estimation of
$\mathbb{E}_{\pi}f$ via subsampling and by using parallel runs instead of a
single run. Our results are applicable to both reversible and non-reversible
Markov chains on discrete as well as general state spaces. We illustrate our
methods by simulations for three examples of Bayesian inference in the context
of risk models and clinical trials.
