Parameter-expanded data augmentation for Bayesian analysis of capture–recapture models

Abstract

Data augmentation (DA) is a flexible tool for analyzing closed and open population models of capture–recapture data, especially models which include sources of hetereogeneity among individuals. The essential concept underlying DA, as we use the term, is based on adding “observations” to create a dataset composed of a known number of individuals. This new (augmented) dataset, which includes the unknown number of individuals N in the population, is then analyzed using a new model that includes a reformulation of the parameter N in the conventional model of the observed (unaugmented) data. In the context of capture–recapture models, we add a set of “all zero” encounter histories which are not, in practice, observable. The model of the augmented dataset is a zero-inflated version of either a binomial or a multinomial base model. Thus, our use of DA provides a general approach for analyzing both closed and open population models of all types. In doing so, this approach provides a unified framework for the analysis of a huge range of models that are treated as unrelated “black boxes” and named procedures in the classical literature. As a practical matter, analysis of the augmented dataset by MCMC is greatly simplified compared to other methods that require specialized algorithms. For example, complex capture–recapture models of an augmented dataset can be fitted with popular MCMC software packages (WinBUGS or JAGS) by providing a concise statement of the model’s assumptions that usually involves only a few lines of pseudocode. In this paper, we review the basic technical concepts of data augmentation, and we provide examples of analyses of closed-population models (M ₀, M _h, distance sampling, and spatial capture–recapture models) and open-population models (Jolly–Seber) with individual effects.

Appendix: Technical conditions for PX-DA

In applications of PX-DA to capture–recapture problems, the conventional (multinomial) model of the complete data must be expanded to account for—and to estimate—the number of non-sampling zeros in the augmented dataset of known size M. The expanded model includes an additional parameter ψ, which is related to zero inflation. Some basic conditions must be satisfied for this expanded model to be innocuous with respect to the original inference problem (Liu and Wu 1999). In this appendix, we show that the expanded model of the augmented data satisfies these conditions.

We begin with a few definitions. Let f correspond to any distribution (posterior, prior, etc.) for the observed data y, and let p denote the same for the augmented data, which we denote by (y, w). (In our applications, w is typically a vector of zeros that correspond to the all-zero capture histories.) Naturally, we desire that the posterior for the augmented data should be equivalent to the posterior based on the observed data. From Liu and Wu (1999), p(N, p | y, w) = f(N, p | y) if and only if p(N, p, ψ) “agrees with” f(N, p) in the following sense:

$$ \int p(N,p,\psi)d\psi = f(N,p) $$

In other words, if the extra parameter ψ is integrated from the prior of the model of the augmented data, this yields the prior for the model of the observed data. In the RDL formulation, this integration yields a $\hbox{U}(0,M) \times \hbox{U}(0,1)$ prior (M being some arbitrarily large integer) for the parameters (N, p), thereby satisfying Liu and Wu’s first condition. The second condition to be satisfied is that the expanded model of the augmented data, p(y, w | N, p, ψ), preserves the model of the observed data, f(y | N, p). This condition is satisfied automatically because our model of the augmented data can be considered as originating from the choice of prior on N (see “PX-DA for Model M0”), not by formulating a model that is structurally distinct from the observed-data model. Therefore, our choice of prior is sufficient to guarantee that we have satisfied the conditions of Liu and Wu (1999) and that the extra parameter ψ used in modeling the augmented data is innocuous to inference about N.