54 
Fishery Bulletin 1 12(1) 
Ay = the stock availability coefficient by period, as- 
sumed to reflect all biological and ecological 
factors affecting the stock; 
= the survey’s global efficiency, assumed to be 
constant and to measure human and techno- 
logical factors of the survey while catching ani- 
mals available in the sampled strata. 
The assumption of constant <P[ implies a time-invari- 
ant sampling protocol, which has broadly prevailed for 
each survey program during the time frame considered. 
Both Ay and modify the survey catchability by pe- 
riod (<7ij ), which was expressed as gy = Ay<£'i. Note that, 
for each survey, q ranges from zero to one; in other 
words, A G [0,1] and <P G [0,1]; q = 0 if A = 0 (unavail- 
able animals during the survey) or <P = 0 (inefficient 
survey). 
Model parameters, derived quantities, and parameter esti- 
mation procedure 
The sets of parameters were 0 = |r,6 oo ,0 r p,a'y , Ay, <£>i, 6^972 j 
for Ml and 0 = jr 0 , a, b^, <7p, erf, Ay, <£1,61972} for M2 - 
The derived metrics included 61972, •••,62008; B„\ -Bi972> 
...,-62008 (assumed to follow a lognormal distribution); 
S6AM61972, . . . ,S6AMP 1989 ; am the expected maximum 
sustainable yield, MSY (MSY = rB,J4); the expected 
biomass and harvest rate at MSY (B^ gy = BJ2 and 
6 msy = r/2); the biomass and fishery-specific harvest 
ratios B\JByi §Y and Hr/HmsyI and for M2, r\ 972, ••• 
> r 2008- The previous metrics for management strictly 
relate to Ml. For M2, Bmsy = 6^/2 and the ratio B t / 
B msy are still valid, but other metrics are year-specific 
(Freon, 1988; for comparable alternatives, see Jacobson 
et al., 2005; Jensen^OOS) and de facto log-linearly re- 
lated to MWET: MSY t = r t Bj4, i^MSYt = u/ 2 > and the 
harvest ratio is 6f t /i?MSYt- Each fishery-specific har- 
vest rate was estimated as Hf t = 6ft/6 t . The total har- 
vest rates and harvest ratios were calculated similarly, 
across fisheries. 
The BSSBDM parameters were assumed to be mu- 
tually independent. The Bayes theorem (e.g., Hilborn 
and Mangel, 1997) was used to estimate the poste- 
rior distributions of the BSSBDM parameters and of 
the derived metrics or statistics of interest. The use 
of the Bayes theorem first required specification of 
prior PDFs, 6(0), about knowledge or hypotheses on 
0 (Table 1), independent of information contained in 
biomass indices. The models were then fitted to the ob- 
served data of biomass indices (O) by using a likelihood 
(or sampling density) function, L(0) = 6(0| 0) and, in 
the process, updated 6(0) into the joint posterior prob- 
ability, 6( 0 | O). 
A prior PDF was developed for the parameter r only 
(Appendix 2) on the basis of Atlantic Croaker demo- 
graphics (Appendix 3). This PDF was applied to both 
Ml and M2 but stood for rq in M2 (Table 1). To aid 
direct comparison of models, priors for other param- 
eters were specified similarly with noninformative dis- 
tributions (here gamma, uniform, or normal; henceforth 
denoted G, U, and N, respectively). Similar to the role 
played by L in lieu of B M , priors were assigned to ay = 
1/Ay and - l/@{ to increase the mixing speed and ef- 
ficiency of the Gibbs sampler underlying BUGS; Ay and 
were derived a posteriori. The choice of noninforma- 
tive priors (Table 1) was dictated by ignorance of most 
parameters, but those priors have been constrained to 
fall within bounds suspected to give support to plau- 
sible parameter values. For example, B r „ was assumed 
to be uniformly distributed between lOx and lOOx the 
observed total fishery removals. 
The Gibbs sampler, a MCMC, numerically inten- 
sive technique implemented in the WinBUGS software 
(vers. 1.4.3; 6 Lunn et al., 2000), was used to sample 
parameter vectors from the joint posterior distribu- 
tions. WinBUGS was run, without starting values, from 
R software (vers. 2.15.3; R Development Core Team, 
2013) by employing the package R2WinBUGS (Sturtz 
et al., 2005). 
The key issue in MCMC simulations is determi- 
nation of when the chain has adequately converged 
(i.e., when the random draws, also called samples, or 
iterations, truly represent the posterior distribution). 
In theory, convergence occurs when the number of it- 
erations increases to infinity, but an infinite number 
of iterations poses problems of computer storage and 
computing time. Moreover, MCMC samples are charac- 
terized by autocorrelation of initial values within the 
chain. In practice and by convention, convergence can 
be achieved by lengthening the chain, autocorrelation 
can be reduced by discarding some initial draws (the 
burn-in period), and disk space is preserved by keeping 
one draw every several iterations (thinning). The burn- 
in period and the thinning interval also must be long. 
In this study, 3 independent chains, each with 
100,000 iterations, a burn-in period of 50,000 draws, 
and a thinning interval of 10 (1 in every 10 values 
was kept) were simulated and led to satisfying conver- 
gence diagnostics. Therefore, 5000 iterations for each 
chain were saved and used for inference. Convergence 
of MCMC simulations to posterior distributions was 
checked by inspecting the traces, autocorrelation plots, 
and Gelman-Rubin (G-R) statistic. In R2WinBUGS, the 
G-R statistic is called a potential scale-reduction factor 
or R statistic; at convergence, R ~ 1, 1.1 being an ac- 
ceptable threshold (Sturtz et al, 2005). This statistic is 
considered sufficient in most practical situations (Rivot 
et al., 2004). The final marginal posterior PDFs were 
summarized in terms of the mean, standard deviation, 
median, and the 2.5th and 97.5th percentiles, which 
define the 95% Bayesian central interval (95% BCI). A 
95% BCI means that there is exactly a 0.95 probability 
that the true value of a parameter lies within that in- 
terval given the model, data, and priors (Ellison, 2004; 
Grobois et al., 2008; Kery, 2010). 
6 Mention of trade names or commercial companies is for iden- 
tification purposes only and does not imply endorsement by 
the National Marine Fisheries Service, NOAA. 
