Appendix B. Algorithms for Gibbs sampling for a growth model. A pdf file of this appendix is also available for viewing.
The conditional posterior for regression parameters contains contributions from the likelihood and prior. For a single unknown variance (Eq. 7, Fig 1. Method 1) this density is
Parameters are sampled directly from , where , , and . The conditional posterior for the variance was sampled from the inverse gamma distribution
For the traditional (ML) method, the model prediction used for s2 and for the second IG parameter is evaluated at .
For Method 2 (Eq. 8, Fig. 1), we sample parameters from the bivariate normal with and . The covariance matrix has plot specific variances along the diagonal. The jth variance has posterior conditional
For the hierarchical model (Eq. 9, Fig. 1, Method 3) regression parameters come from , where , , , and . Hyperparameters are drawn from , where and . The parameter covariance matrix is drawn from the Wishart conditional posterior
(e.g., Gelman et al. 1995). Although regression parameters are given in matrix notation (for compactness), matrix inversions are too slow. All parameters are solved without resort to matrix inversion or (for the hierarchical model) loops over individuals.
Half saturation constant
Parameter depends on the likelihood and on its prior Beta density. For Method 1, the conditional posterior is
with appropriate modifications to the first distribution for different models. We used a Metropolis-Hastings step with a beta proposal density centered on the current value of .
Latent (“true”) light availabilities
Light availability on plot j enters the likelihood for the nj seedlings on plot j, and it has a density describing measurement uncertainty. It is conditionally independent of other plots,
Sampling was executed with a Metropolis-Hastings step based on a Beta proposal density.