Appendix E. Bayesian regressions of individual deviations against traits.
The full posterior distribution of the model given by Eq. (3) is given by
where with is identity matrix and is the dimension of vector , and is the marginal posterior distribution of the individual deviations in capture probabilities, which is obtained from the main model, and used here as a prior distribution. For the parameters , , and we used a non-informative prior distribution . The full conditional posterior distributions used in the Gibbs sampling are given as follows.
1. The full conditional posterior distribution of is given by
The ’s were sampled using the Metropolis-Hastings algorithm with proposal distribution .
2. The full conditional posterior distribution of the parameters , , and is given by
These parameters were sampled using a standard procedure (Gelman et al. 2004), which involves two steps. (i) is sampled from the marginal posterior distribution , and (ii) and are sampled from the full conditional posterior distribution. Here , , and are given by
Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin. 2004. Bayesian data analysis. Chapman and Hall/CRC.