  This article provides the first procedure for computing a fully
data-dependent interval that traps the mixing time $t_{\text{mix}}$ of a finite
reversible ergodic Markov chain at a prescribed confidence level. The interval
is computed from a single finite-length sample path from the Markov chain, and
does not require the knowledge of any parameters of the chain. This stands in
contrast to previous approaches, which either only provide point estimates, or
require a reset mechanism, or additional prior knowledge. The interval is
constructed around the relaxation time $t_{\text{relax}}$, which is strongly
related to the mixing time, and the width of the interval converges to zero
roughly at a $\sqrt{n}$ rate, where $n$ is the length of the sample path. Upper
and lower bounds are given on the number of samples required to achieve
constant-factor multiplicative accuracy. The lower bounds indicate that, unless
further restrictions are placed on the chain, no procedure can achieve this
accuracy level before seeing each state at least $\Omega(t_{\text{relax}})$
times on the average. Finally, future directions of research are identified.
