  We derive general bounds on the probability that the empirical first-passage
time $\overline{\tau}_n\equiv \sum_{i=1}^n\tau_i/n$ of a reversible ergodic
Markov process inferred from a sample of $n$ independent realizations deviates
from the true mean first-passage time by more than any given amount in either
direction. We construct non-asymptotic confidence intervals that hold in the
elusive small-sample regime and thus fill the gap between asymptotic methods
and the Bayesian approach that is known to be sensitive to prior belief and
tends to underestimate uncertainty in the small-sample setting. We prove sharp
bounds on extreme first-passage times that control uncertainty even in cases
where the mean alone does not sufficiently characterize the statistics. Our
concentration-of-measure-based results allow for model-free error control and
reliable error estimation in kinetic inference, and are thus important for the
analysis of experimental and simulation data in the presence of limited
sampling.
