M

These can be converted to per capita birth and death rates by appropriate normalization. The superpopulation size under the occupancy derived parameterization is model {

gamma[T]~dunif(0,1) p[1]<- 1; p[T]<- 1 for(i in 2:(T-1)){ p[i]~dunif(0,1)

z[i,1]~dbern(gamma[1]) mu[i]<-z[i,1]*p[1] y[i,1]~dbern(mu[i]) recruitable[i,1]<-1 for(j in 2:T){

recruitable[i,j]<- recruitable[i,(j-1)]*(1-z[i,j-1])

mu2[i,j]<- survived[i,j] + gamma[j]*recruitable[i,j]

Panel 10.1. WinBUGS model specification of the Jolly-Seber type model as a constrained model of patch occupancy. Here, M is the size of the list, including n observed individuals and M-n augmented all-zero encounter histories. The M x T data matrix is y.

Here, I(arg) is an indicator function that evaluates to 1 if individual i was ever alive.

Within an MCMC framework for inference, in which the missing values of z(i,t) are updated, these derived parameters can be computed at each iteration of the algorithm and their posterior distributions estimated by the resulting collection of simulated values. These derived quantities have metapopulation analogs, which we referred to as finite-sample quantities in Chapter 9 (see also Royle and Kery (2007)).

10.3.6 Analysis of the European Dipper Data

Here we provide an analysis of the classical European dipper data described by Lebreton et al. (1992). These data are from a 7-year study of European dippers (Cinclus cinclus), originating from Marzolin (1988), and have been used extensively by others including Brooks et al. (2000) and Royle (2008c). The number of unique individuals first captured in each of the 7 periods was (22, 60, 78, 80, 88, 98, 93). These data appear to be widely available, e.g., from E.G. Cooch's website: http: //www.phidot.org/software/mark/docs/book/.

Bayesian estimates under the occupancy parameterization are given in Table 10.1. The columns labeled 'prior 1' were obtained under the specification Yt — U(0,1), as well as uniform priors for the remaining probability parameters. Estimates in Table 10.1 include both the canonical 'colonization' probabilities, Yt, and also the derived entrance probabilities nt of the SA parameterization (which we address later).

10.3.7 Prior Distributions for Recruitment

In Section 10.4 we provide estimates of N using alternative parameterizations (and also the MLEs). We will note general inconsistencies between estimates of N using different parameterizations of the model, as well as between the MLEs and the Bayesian estimates. In small samples such inconsistencies are typical in multi-parameter models because posterior means and multi-dimensional modes are different quantities. However, another important consideration has to do with prior specification. Since all of the parameters of the model are probabilities, we used the conventional U(0,1) priors for all parameters. This choice of priors is not completely innocuous. Note that

E(N|M, Y1, Yt,..., Yt) = M {1 — (1 — Y1)(1 — Y2)... (1 — Yt)} .

As such, when we tinker around with priors on Yt, we are inducing prior structure on N.

Data augmentation was motivated (Section 10.1) as arising under the assumption of a discrete uniform prior for N on the integers [0, M], which was constructed hierarchically as a Bin(M, ft) for N and a U(0,1) prior for ft. However, ft is not specified directly in the occupancy-based parameterization, instead it is a derived parameter. The model is constructed in terms of the recruitment parameters Yt. Under the occupancy-derived parameterization, jt ~ U(0,1). The result is that the implied prior for the inclusion probability, i.e., for ft = 1 — (1-Yi)(1-y2)...(1-yt), is not U(0,1). As such, the resulting implied prior on N is not uniform on the integers [0, M] as desired to justify data augmentation (see Section 5.6). In addition, the implied entrance probabilities are not uniform across the T periods. These facts turn out to have some influence on the posterior of N and, consequently, some of the other parameters as well.

One way to remedy this potential sensitivity to prior distributions is to parameterize the model directly in terms of ft - place a uniform prior on that parameter, and specify equal entrance probabilities across the T periods (i.e., place a Dirichlet prior having sample sizes at = 1). This solution arises naturally as the data augmentation version of the SA model (Section 10.4). Alternatively, we can experiment with the priors Yt ~ Beta(at,bt) such that the fraction of individuals ever recruited (out of M) is 0.5, and an equal number (in expectation) recruit at each time period. For the dipper data, T = 7 periods, we would like a prior that allocates 1/14 of the individuals to each period, so that 7/14 of the individuals on the augmented list are members of the superpopulation. One way to achieve this is to set at = S/(14 — (t — 1)) and bt = S — at, where S « 1.5 seems to yield approximately a U(0,1) prior for ft. These results are shown under 'prior 2' in Table 10.1. Compared to 'prior 1' in Table 10.1, we see a slight difference between the posterior of N as well as the other parameter estimates. We provide context for these estimates in Section 10.4.2. The WinBUGS model specification for this parameterization, and its implementation in R, are provided on the Web Supplement.

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