Inference from a Deterministic Population Dynamics Model for Bowhead Whales

Abstract

We consider the problem of inference about a quantity of interest given different sources of information linked by a deterministic population dynamics model. Our approach consists of translating all the available information into a joint premodel distribution on all the model inputs and outputs and then restricting this to the submanifold defined by the model to obtain the joint postmodel distribution. Marginalizing this yields inference, conditional on the model, about quantities of interest, which can be functions of model inputs, model outputs, or both. Samples from the postmodel distribution are obtained by importance sampling and Rubin's SIR algorithm. The framework includes as a special case the situation where the pre-model information about the outputs consists of measurements with error; this reduces to standard Bayesian inference. The results are in the form of a sample from the postmodel distribution and so can be examined using the full range of exploratory data analysis techniques. Methods for comparing competing population dynamics models are developed, based on a generalization of the Bayes factor idea. A key quantity used by the International Whaling Commission (IWC) in making decisions about bowhead whales, Balaena mysticetus, is the replacement yield, RY. Information about the species is of three main types: recent census information, historical catch records, and evidence about birth and death rates. These are combined using a special case of the Leslie matrix population dynamics model. Our method yields full inference about RY and also sheds light on other, sometimes controversial, questions of scientific interest. These ideas are also applicable to many simulation models in other areas of science and policy making. Software to implement these methods is available from StatLib.