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Fishery Bulletin 105(4) 



distributions of the movement parameters for any of 

 the six alternate sets of parameter values considered 

 (Fig. 5). A further check was made by rerunning the 

 model assuming high (75%) initial tag loss; again no 

 appreciable change in posterior distributions was ob- 

 served, and median parameter values changed by less 

 than 0.001. 



The ability to quantify the plausibility of the observed 

 data given the fitted model, Bayesian goodness-of-fit, 

 was examined by checking the posterior predictive dis- 

 tribution (the probability distribution for an unobserved 

 data point), which can indicate the degree to which the 

 model structure, priors, and likelihoods assumed in 

 the model are appropriate (Gelman et al., 1995). The 

 posterior predictive distribution for expected recover- 

 ies corresponding to each of the observed recoveries 

 was generated during MCMC sampling. The posterior 

 predictive check compares the observed data to a distri- 

 bution of predictions and summarizes the information 

 across data types. The mean standardized residuals 

 were calculated by dividing the raw residual (between 

 the observed value and the j:* percentile of the poste- 

 rior predictive distribution) by the expected standard 

 deviation (based on the negative binomial likelihood), 

 and by taking the mean of these values for each number 

 of observed recoveries. Figure 6 shows the mean stan- 

 dardized residual for the 95"^, 50*, and 25'^ percentiles 

 of the posterior distribution of expected recoveries for 



each of the four types of data. There are a few obser- 

 vations well in excess of the range expected for stan- 

 dardized residuals, primarily for the type-1 data sets. 

 In addition, some trend was observed in the residuals 

 for the type-4 data sets with larger residuals occurring 

 at the largest observed values. Further, although the 

 zero-line for the 5"^ percentile of the predictions does 

 lie below more than 95% of the residuals, there appears 

 to be an excess of residuals above the zero-line for the 

 95th percentile of the posterior predictions. This excess 

 of residuals indicates that model predictions generally 

 resulted in fewer recoveries (for some time and space 

 combinations) than were observed. 



Discussion 



Bayesian analyses are ideal for fisheries applications 

 because uncertainty is explicitly and transparently 

 incorporated into them, they allow for the use of sev- 

 eral data sources (Hilborn and Mangel, 1997), provide 

 easily interpretable probability inference (Wade, 2000), 

 and yield results suitable for formal decision analysis 

 (McAllister et al., 1994). The Bayesian framework devel- 

 oped here allows calculation of probability distributions 

 for key parameters governing English sole movement 

 rates. The results from this analysis qualitatively sup- 

 port what can be directly inferred from the original 



