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

TL;DR

This paper develops non-asymptotic bounds and confidence intervals for empirical first-passage times in reversible ergodic Markov processes, enabling reliable uncertainty quantification in small-sample scenarios.

Contribution

It introduces sharp, model-free bounds and confidence intervals for empirical first-passage times, addressing the small-sample regime gap in uncertainty quantification.

Findings

Provides non-asymptotic confidence intervals for small samples

Derives sharp bounds on extreme first-passage times

Enables reliable error estimation in kinetic inference

Abstract

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.