Inference of Transition Rates in a Birth-Death Chain from Conditional Exit Times

Consider a birth-death process (BDP) of length N + 1 with general birth-death rates, which has a maximum population of N and becomes extinct when the population reaches zero. In this talk, a method of recovering the birth-death rates of the BDP from its extinction times (ETs) is presented. Given that the maximum site n reached by each trajectory is also known, we use the proportion of trajectories that do not exceed n and corresponding mean ET to recover the birth-death rates. Our method is based on properties of the characteristic polynomial of the BDP's infinitesimal generator. Given 50 million ETs of an 11-site birth-death chain, we can recover all the rates with a relative error about 3%. Our method could be used to infer potential landscape information for single-molecule spectroscopy experiments, or to compute the mutation rates of cancer cells undergoing sequential genotypic changes.