Wald’s martingale and the Moran process

Abstract Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moran birth-death process, where that quantity is the number of mutants in a population. In such processes we are often interested in their absorption (i.e. fixation) probabilities, and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth-death model. Instead of considering the time to absorption, we consider a closely-related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical, and exact. The parameter dependence of the characteristic functions is explicit, so it is easy to explore their properties in parameter space. We also use them to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so we can quickly and tractably obtain absorption probabilities and times of evolutionary stochastic processes. Author summary The Moran process is a probabilistic birth-death model of evolution. A mutant is introduced to an indigenous population, and we randomly choose organisms to live or die on subsequent time steps. Our goals are to calculate the probabilities that the mutant eventually dominates the population or goes extinct, and the distribution of time it requires to do so. The conditional distributions of time are difficult to obtain for the Moran process, so we consider a slightly different but related problem. We instead calculate the conditional distributions of the number of times that the mutant population size changes before it dominates the population or goes extinct. We use a martingale identified by Abraham Wald to obtain elegant and exact expressions for those distributions. We then use them to approximate conditional time distributions, and we show when that approximation is accurate. Our analysis outlines the basic concepts martingales and demonstrates why they are a formidable tool for studying probabilistic evolutionary models such as the Moran process.