The joint posterior distribution was determined by Gibbs sampling (Gelfand and Smith 1990). A Gibbs sampler is a Markov chain Monte Carlo technique that executes a random walk with a limiting distribution equal to the parameter distribution of interest. Because random samples cannot be drawn directly from this complex distribution, we factor the posterior into conditional distributions given here. We used 10,000 iterations starting from different initial values. A ‘burn in’ of 500 steps was discarded. Ninety-five percent Bayesian credible intervals are posterior quantiles. Propagation to the response m was determined using quantiles of m drawn from the 9500 MCMC samples for all parameters. 

Regression parameters

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 j^th 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 n_j 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.

Literature cited

Gelfand, A. E., and A. F. M. Smith. 1990. Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85:398–409.

Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin. 1995. Bayesian data analysis. Chapman and Hall, London, UK.