Bayesian population models can be exceedingly slow due, in part, to the choice to simulate discrete latent states. Here, we discuss an alternative approach to discrete latent states, marginalization, that forms the basis of maximum likelihood population models and is much faster. Our manuscript has two goals: 1) to introduce readers unfamiliar with marginalization to the concept and provide worked examples, and 2) to address topics associated with marginalization that have not been previously synthesized and are relevant to both Bayesian and maximum likelihood models. We begin by explaining marginalization using a Cormack-Jolly-Seber model. Next, we apply marginalization to multistate capture-recapture, community occupancy, and integrated population models and briefly discuss random effects, priors, and pseudo-R2. Then, we focus on recovery of discrete latent states, defining different types of conditional probabilities and showing how quantities such as population abundance or species richness can be estimated in marginalized code. Lastly, we show that occupancy and site abundance models with auto-covariates can be fit with marginalized code with minimal impact on parameter estimates.
Marginalized code was anywhere from five to >1000 times faster than discrete code. Differences in inferences were minimal using marginalized code. Discrete latent states and fully conditional approaches provide the best estimates of conditional probabilities for a given site or individual. However, estimates for parameters and derived quantities such as species richness and abundance were minimally affected by marginalization and use of imperfect estimates of conditional probabilities. The results applied even when auto-covariates based on imperfect estimates of conditional probabilities were used. Understanding how marginalization works shrinks the divide between Bayesian and maximum likelihood approaches to population models. Some models that have only been presented in a Bayesian framework can easily be fit in maximum likelihood. On the other hand, factors such as informative priors, random effects, or pseudo-R2 values may motivate a Bayesian approach in some applications. An understanding of marginalization allows users to minimize the speed that is sacrificed when switching from a maximum likelihood approach. Widespread application of marginalization in Bayesian population models will facilitate more thorough simulation studies, comparisons of alternative model structures, and faster learning.