Appendix C. The Bayesian approach.
A full probability model that consists of a joint probability distribution for all observable and unobservable quantities must be constructed. A full joint-probability distribution is specified by assigning conditional probability distributions for each quantity in the model that is related to other parameters or observed data. In addition, prior probability distributions must be assigned to those parameters which are not directly conditioned on other parameters or data . In this step the Bayesian approach allows the incorporation of prior knowledge that might be available (Ellison 1996). If only uniform priors are used, Bayesian posterior modes (equivalent to posterior means for symmetrical distributions) will be equal to and have the same performance as (in the language of a frequentist) the maximum likelihood estimates (Berger 1985).
We performed 300 000 iterations of the Gibbs sampler, WinBUGS (the BUGS version for Windows available at http://www.mrc-bsu.cam.ac.uk/bugs/) version 1.3, after a “burn-in” of 50 000 iterations. The chain was thinned by taking every 15th observation (this keeps the output file down to a reasonable size). It is necessary to check that the Markov chain has been run for sufficiently long to ensure that it is sampling from its equilibrium distribution. Additionally, it must be checked whether the chain "mixes" well. This ensures that the chain is able to move easily through the whole space of the posterior distribution.
Convergence diagnostics were checked using BOA (Bayesian Output Analysis Program, available at http://www.public-health.uiowa.edu/boa/), which is based on the convergence and output analysis software of Best et al. (1995). For all prairie-parkland areas and both species the diagnostics suggested full convergence. More specifically, when two parallel chains were started from different over-dispersed initial values, they converged to the same equilibrium distribution (the test of Gelman and Rubin 1992). Most chains passed the Heidelberger and Welch (1983) stationarity and halfwidth test. Autocorrelations within each chain were reasonable low and suggested that the "mixing" of chains were adequate. Additionally, only minor changes in estimates were found when preliminary analyses of longer runs were made.
For references and convergence diagnostics see (http://www.mrc-bsu.cam.ac.uk/bugs/): Best et al. (1995).
Berger, J. O. 1985. Statistical decision theory and Bayesian analysis. Second Edition. Springer-Verlag, New York, New York, USA.
Best, N. G., M. K. Cowles, and S. K. Vines. 1995. CODA Convergence diagnosis and output analysis software for Gibbs sampler output: version 0.3. MRC Biostatistics Unit, Institute of Public Health, Cambridge, UK.
Ellison, A. M. 1996. An introduction to Bayesian inference for ecological research and environmental decision-making. Ecological Applications 6:10361046.
Gelman, A., J. B Carlin, H. S. Stern, and D. B. Rubin. 1995. Bayesian data analysis. Chapman and Hall, London, UK.
Heidelberger, P., and P. D. Welch. 1983. Simulation Run Length Control in the Presence of an Initial Transient. Operations Research 31:11091144.