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We derive general bounds on the probability that the empirical first-passagetime $\overline{\tau}_n\equiv \sum_{i=1}^n\tau_i/n$ of a reversible ergodicMarkov process inferred from a sample of $n$ independent realizations deviatesfrom the true mean first-passage time by more than any given amount in eitherdirection. We construct non-asymptotic confidence intervals that hold in theelusive small-sample regime and thus fill the gap between asymptotic methodsand the Bayesian approach that is known to be sensitive to prior belief andtends to underestimate uncertainty in the small-sample setting. We prove sharpbounds on extreme first-passage times that control uncertainty even in caseswhere the mean alone does not sufficiently characterize the statistics. Ourconcentration-of-measure-based results allow for model-free error control andreliable error estimation in kinetic inference, and are thus important for theanalysis of experimental and simulation data in the presence of limitedsampling.

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime