Optimal sampling and problematic likelihood functions in a simple population model

Abstract

Markov chains provide excellent statistical models for studying many natural phenomena that evolve with time. One particular class of continuous-time Markov chain, called birth-death processes, can be used for modelling population dynamics in fields such as ecology and microbiology. The challenge for the practitioner when fitting these models is to take measurements of a population size over time in order to estimate the model parameters, such as per capita birth and death rates. In many biological contexts, it is impractical to follow the fate of each individual in a population continuously in time, so the researcher is often limited to a fixed number of measurements of population size over the duration of the study. We show that for a simple birth-death process, with positive Malthusian growth rate, subject to common practical constraints (such as the number of samples and timeframes), there is an optimal schedule for measuring the population size that minimises the expected confidence region of the parameter estimates. This type of experimental design results in a more efficient use of experimental resources, which is often an important consideration