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.

翻译：我们推导出了可逆遍历马尔可夫过程经验首达时间$\overline{\tau}_n\equiv \sum_{i=1}^n\tau_i/n$（基于$n$个独立实现的样本）偏离真实平均首达时间任意给定方向程度概率的通用界。我们在难以处理的有限样本条件下构建了非渐近置信区间，从而填补了渐近方法与贝叶斯方法之间的空白——后者对先验信念敏感且易在有限样本场景中低估不确定性。我们证明了极端首达时间的尖锐界限，即使在均值无法充分表征统计特性的情形下也能有效控制不确定性。基于测度集中方法所获得的成果，可实现动力学推断中无模型误差控制与可靠误差估计，这对分析实验及模拟数据中的采样受限问题具有重要价值。