Quantifying uncertainty in demographic models: Bayesian methods for Integral Projection Models (IPMs)
Integral projection models (IPMs) are a powerful and popular approach to modeling demographic data. We use hierarchical Bayesian methods to construct a stochastic IPM for a long-lived plant species, the cholla (Opuntia imbricata), which offers distinct advantages over other approaches given the often hierarchical nature of demographic data. We highlight the advantages of a hierarchical Bayesian approach for modeling demographic data and translating uncertainty in the vital rates to uncertainty in population-level quantities derived from them.
The best-fit demographic model, which would have been difficult to fit under a standard frequentist framework, allowed for spatio-temporal differences in growth, survival, and fertility, and correlated responses to temporal variation across vital rates. The corresponding posterior probability distribution for the stochastic population growth rate (λs) indicated that, if current vital rates continue, the study population would decline with nearly 100\% probability. While similar models may be fit using a frequentist or maximum likelihood framework, they are not, from our perspective, as intuitive. From a computational perspective, the advantages of hierarchical Bayesian approaches to IPM modeling include that (1) they provide a natural fit to demographic data, which are often inherently hierarchical; (2) they allow investigators to seamlessly combine multiple data sets or experiments within a single analysis; (3) they can readily incorporate covariance between vital rates; and, (4) they allow investigators to easily integrate prior information from related taxa, which may be particularly important for species of conservation concern. However, constructing a hierarchical Bayesian model will often require the custom development of a statistical model associated with the peculiarities of the sampling design and species considered. Even though Bayesian methods are not required for IPM modeling, we suggest that they will often be preferable given the weight of advantages over disadvantages. Overall, Bayesian approaches provide a statistically sound way to get more information out of hard-won data, the goal of most research endeavors.