The importance of prior choice in model selection: a density dependence example

It is important to discern the magnitude of density dependence a species exhibits, as well as the time lag over which it operates. Knowledge of a species' likely response to natural as well as synthetic shocks will assist in effective species management. Statistically this is a challenging problem which does not usually admit closed-form mathematical analysis. Consequently, many people have used Bayesian methods to fit state space models of density dependence to many different species, of which we take eleven species of North American duck as our motivating examples. A Bayesian analysis requires a choice of model and parameter prior. The latter is difficult to do without inducing bias in model selection, and we attempt to address this problem. Our priors will be obtained by considering which parameter values are representative of features we expect to see in the data, and which would produce unnatural behaviour. To fit the models, we use a novel sequential Monte Carlo method (particle learning) not previously applied to ecological data sets. We show that existing analyses on the duck data may have been susceptible to a common problem in Bayesian model selection (Lindley's paradox), and suggest methods for prior selection which mitigate this issue. We also discover that although it is possible to detect the existence of density dependence, it is unrealistic to expect to determine the time lag over which it operates without a great deal of data, even if said data are simulated from the model. We demonstrate that prior choices motivated by the above considerations can lead to substantially increased predictive accuracy over surprisingly long time scales whether model selection is of primary concern or not. We conclude from our analysis of real-world data that there is little evidence of density dependence in many duck species, suggesting that such effects, if present, are likely to be small in magnitude.