## Info

is a model for the binary observations y(i,t) conditional on the latent state variables z(i,t).

### 11.3.0.2 The state model

Let z(i, t) for i = 1,..., n and t = 1,..., T be Bernoulli random variables describing whether individual i is alive (z(i,t) = 1) or dead (z(i,t) = 0) at time t. The CJS model is developed conditional on the time of first capture of each individual, fj. That is, z(i,fj) = 1 with probability 1. The survival process is given by the conditional model for t = fj + 1,..., T. In other words, if an individual is alive at time t — 1, its survival outcome is a Bernoulli random variable with parameter ^t-1. If an individual is not alive at t — 1, then z(i,t) is Bernoulli with success probability 0, i.e., z(i,t) = 0 with probability 1.

11.3.0.3 The observation model

Conditional on the state process, y(i, t) are independent Bernoulli random variables,

Thus, if z(i,t) = 0 then y(i,t) = 0 with probability 1, otherwise y(i,t) is a Bernoulli trial with parameter pt.

The models have been described here with time-varying parameters. While this is a common model, so, too, are age-structured models, or models in which either pt or vary in response to known covariates. These and other modifications are both straightforward to construct and practical to implement in WinBUGS as we describe subsequently.

### 11.3.1 Bayesian Analysis

The hierarchical formulation of the CJS model is easily described in WinBUGS pseudo-code as shown in Panel 11.4 for a simple case in which both p and ^ are constant. Here, the data are the encounter history matrix y, the vector first and the fixed constants nind and nyear. The line z[i,first[i]] — dbern(1) admits that the first latent state variable of each individual is fixed, whereas the meaning of the remaining model specification should be self-evident. In particular, note that z(i,t)|z(i,t — 1) — Bern(z(i,t — 1)^t-1)

U(0,1) priors are assumed for both ^ and p, and the distributions of z(i,t)|z(i,t — 1) and y(i,t)lz(i,t) are those described by Eqs. (11.3.1) and (11.3.2), respectively.

We illustrate the fitting of this model using WinBUGS with a small data set shown in Table 11.3. These data are encounter histories of 10 yellow warblers (Dendroica petechia) obtained from a constant-effort mist netting station that is part of the Monitoring Avian Productivity and Survival (MAPS) program (see Section 11.4 for additional details). These individuals were first captured in 1992 and the data include recaptures through 2000. The model described in Panel 11.4 was fitted to these data (an R script is provided in the Web Supplement). Posterior summary statistics for the warbler data are given in Table 11.4. These summaries are based on 60000 Monte Carlo draws after discarding 2000 burn-in samples.

11.3.2 Representation as a Constrained Jolly-Seber Model

In Chapter 9 we described the related class of models known as Jolly-Seber models which allow for both recruitment and survival. In the CJS model, by conditioning model {

## Prior distributions phi~dunif(0,1)

### Individuals enter sample with probability 1

z[i,first[i]]~dbern(1) ### Definition of state and observation models for(j in (first[i]+1):nyear){ mu1[i,j]<-p*z[i,j] y[i,j]~dbern(mu1[i,j]) mu2[i,j]<-phi*z[i,j-1] z[i,j]~dbern(mu2[i,j])

Panel 11.4. WinBUGS model specification for fitting the CJS model with constant p and 0.

Table 11.3. Yellow warbler (Dendroica petechia) encounter history data for a sample of 10 individuals from a constant-effort mist netting station. Individuals were all first captured in 1992.

t= 1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 1 0 0 0 0 0 0 0 0~~ 100000000 100000000 1 1 1 1 0 0 0 0 0 110 0 11111 100000000 100000000 100000000 100000000 100000000

Table 11.4. Posterior summary statistics from analysis of the warbler data (De'ndroica petechia) set under a constant-p, constant-0 model. These data are all individuals encountered at a single MAPS station over a nine year period.

parameter mean SD q0.025 90.50 90.975 0 0.571 0.105 0.361 0.573 0.767

on capture, or entry into the sample, we lose information about recruitment parameters. The relationship between the two models can be seen by considering the Jolly-Seber state model, as described in Chapter 9, which is:

z(i,t + 1) - Bern ^<Mi,t) + Yt+1 jll(1 — z(i,k))

where k=1(1 — z(i, k)) is an indicator of previous recruitment. That is, an individual can be recruited with probability Yt+1 if it has never previously been recruited. Evidently the CJS model is a formal reduced version of this model, which we obtain by setting Yt = 0. In addition, entry into the sample is regarded as a random variable under the Jolly-Seber model. Thus, the observation model will contain additional probability structure for the initial encounter, such as y(i, 1) — Bern(p), for each i = 1, 2,..., n.