*Ecological Archives* E095-029-A1

James T. Thorson, Kotaro Ono, Stephan B. Munch. 2014. A Bayesian approach to identifying and compensating for model misspecification in population models. *Ecology* 95:329–341. http://dx.doi.org/10.1890/13-0187.1

Appendix A. Markov chain Monte Carlo methods.

We used Metropolis within Gibbs sampling (Clark and Bjørnstad 2004), i.e., we cycled sequentially through all model parameters and, for each, sampled a proposal value from a normal distribution that was accepted or rejected based on the ratio of posteriors for the old and proposed values. Sampling chains for the conventional surplus production model were started at randomized locations, while chains for the GP model were started at mean estimates from the conventional model for that simulation replicate. All parameters were sampled in log-space, except for *θ* which was sampled in logit-space, and exploratory analysis confirmed that sampling of ln(*B _{t}*) resulted in low sampling correlations for these state variables. Sampling used three chains, each with 10,000 burn-in, and burn-in samples were used to adapt the Metropolis proposal kernel to achieve an acceptance rate between 20–50%. We then monitored each chain for 10,000 samples, with a thinning rate of 50 (i.e., 600 retained samples total). Models were implemented in the REPORT_SECTION of ADMB (Fournier et al. 2012), called from within R (R Development Core Team 2011), and chains were parallelized using the "snowfall" package (Knaus 2010).

We checked for evidence of non-convergence using conventional methods, i.e., trace plots of sampled parameters, autocorrelation plots, the Gelman-Rubin R statistic, and estimates of the effective sample size using the package "coda" (Plummer et al. 2009). The highest R-statistic for each run was almost always <1.05, which in most cases corresponds to an effective sample size >50. This value of effective sample size implies that further sampling would result in a small but non-zero decrease the sampling imprecision due to the numeric approximation of marginal posteriors. We confirmed that longer MCMC samples had no noticeable effect on results, and used these chain lengths to allow for a greater number of simulation replicates and scenarios.

Literature cited

Clark, J. S., and O. N. Bjørnstad. 2004. Population time series: process variability, observation errors, missing values, lags, and hidden states. Ecology 85:3140–3150.

Fournier, D. A., H. J. Skaug, J. Ancheta, J. Ianelli, A. Magnusson, M. N. Maunder, A. Nielsen, and J. Sibert. 2012. AD Model Builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models. Optimization Methods and Software 27:1–17.

Knaus, J. 2010. snowfall: Easier cluster computing (based on snow).

Plummer, M., N. Best, K. Cowles, and K. Vines. 2009. coda: Output analysis and diagnostics for MCMC.

R Development Core Team. 2011. R: A Language and Environment for Statistical Computing. Vienna, Austria.