Concentration of Empirical First-Passage Times

TL;DR

This paper discusses recent advances in understanding the variability of empirical first-passage times in reversible Markov processes, providing concentration inequalities and confidence intervals to improve estimation from limited data.

Contribution

It introduces new concentration bounds and confidence intervals for empirical first-passage times, addressing finite-sample effects and uncertainty quantification.

Findings

Derived concentration inequalities for empirical first-passage times.

Constructed non-asymptotic confidence intervals for mean estimates.

Provided bounds on fluctuations of maximum and minimum first-passage times.

Abstract

First-passage properties are central to the kinetics of target-search processes. Theoretical approaches so far primarily focused on predicting first-passage statistics for a given process or model. In practice, however, one faces the reverse problem of inferring first-passage statistics from, typically sub-sampled, experimental or simulation data. Obtaining trustworthy estimates from under-sampled data and unknown underlying dynamics remains a daunting task, and the assessment of the uncertainty is imperative. In this chapter, we highlight recent progress in understanding and controlling finite-sample effects in empirical first-passage times of reversible Markov processes. Precisely, we present concentration inequalities bounding from above the deviations of the sample mean for any sample size from the true mean first-passage time and construct non-asymptotic confidence intervals. Moreover, we present two-sided bounds on the range of fluctuations, i.e, deviations of the expected maximum and minimum from the mean in any given sample, which control uncertainty even in situations where the mean is a priori not a sufficient statistic.