92 



Fishery Bulletin 104(1) 



However, the fishing mortality rate that maximizes 

 Equation 12 also maximizes Equation 12 divided by the 

 virgir. recruitment /fg (a constant). Thus, i^^v/sv ™3y be 

 obtained from 



max 



F 





e ■-" 



(13) 



where s Jp has been substituted for RplR^y 



The values of p and s corresponding to F„,^,^ , Fq ,, 

 or F,„,,. may be calculated by means of Equations 9 and 

 10, respectively. Note however, that p is no longer the 

 target value specified by management, but a derivative 

 of the targeted values of F. This means that MSST defi- 

 nitions based on s, ,, , s,, ,, and s,., will vary somewhat 



III (l\ '.I 1 I'lsv -^ 



with the behavior of the fishery. In some cases this 

 could lead to risk prone situations where the percep- 

 tion of stock status changes simply because the fishery 

 targets different age groups (i.e., the definition of MSST 

 changes rather than the abundance of the resource). In 

 the case of MSY. a more stable alternative is to define 

 the MSST in terms of a "spawn at least once" policy 

 where mature animals are regarded as fully vulnerable 

 to the fishery and immature animals area regarded as 

 invulnerable. 



Parameter estimation 



The equations above include numerous "unknowns" rep- 

 resenting the processes of reproduction, mortality, and 

 growth. In the case of "data-poor" stocks like goliath 

 grouper, there are insufficient data to estimate all of 

 these unknown parameters with an acceptable level of 

 precision. However, it is often possible to increase the 

 precision of the estimates through the use of Bayesian 

 prior probability densities constructed to reflect expert 

 opinion (e.g., Wolfson et al., 1996; Punt and Walker, 

 1998) or based on meta-analyses involving similar spe- 

 cies (e.g., Liermann and Hilborn, 1997; Maunder and 

 Deriso, 2003). 



The Bayesian approach to estimation seeks to develop 

 a "posterior" probability density for the parameters 

 that is conditioned on the data D, P{0 I D). By applica- 

 tion of Bayes rule it is easy to show that 



P{0\D)c<P{D\0}P{0). 



(14) 



where P(D\0) is the sampling density (likelihood func- 

 tion); and P(0) is the prior density (in this case the 

 analyst's best guess of the probability density for 0). 



Estimates for may be obtained by integrating the 

 posterior (the classical Bayes moment estimator; cf. 

 Gelman et al., 1995) 



Oj = \e^P{D\0)P(0)dej, 0, 6 



(15) 



or by minimizing its negative logarithm (the highest 

 posterior density estimator; Bard, 1974) 



min{-log.P(I>l0)-log,P(0)|. 



(16) 



In the present model, a prior needs to be specified 

 for the parameters reflecting recruitment (o and £ ), 

 mortality (M, (p. d^,, i',, ), fecundity (£„', and growth in 

 weight ((f„). It is assumed in the present study that the 

 parameters are statistically independent with respect 

 to prior knowledge, such that the joint prior is merely 

 the product of the marginal priors for each parameter. 

 The exceptions are the process error functions for the 

 annual recruitment and fishing mortality rate devia- 

 tions, f^ and (3^. These are assumed to be autocorrelated 

 lognormal variates with negative- log density functions 

 of the form 



-logP(f) = 



2a 



fr + 



I*' 



-pr£.y 



- log cr^. 



(17) 



where p, = the correlation coefficient; and 

 o'\ = the variance of log^.j;^. 



For stability reasons, it is assumed that £g = 0. 



It is possible, at least in principle, to conduct an as- 

 sessment based on prior specifications alone. However, it 

 may be difficult to develop sufficiently informative pri- 

 ors for some of the parameters, particularly for the fish- 

 ing mortality rates. The preferred approach, of course, 

 is to condition the estimates on data. With the present 

 model it is assumed that catch data are either unavail- 

 able or unreliable, otherwise a standard age-structured 

 production model (cf. Restrepo and Legault, 1998) would 

 be more appropriate. However, time series of catch-per- 

 unit-of-effort data or fishery-independent surveys are 

 often available even when total catches are not. To the 

 extent that changes in these data (r) are proportional to 

 changes in the abundance of the population as a whole 

 (N), they may be modeled as 



C,..=Q,Y.^,,,Na.ye 



-(f,r„+M„l(, y,^^ 



(18) 



y, ^, - NonnahO.a^., ), 



where / 



t. = 



and 



ndexes the particular survey time series; 



= the proportionality coefficient scaling the 

 time series to the relative abundance of the 

 population; 



the fraction of the year elapsed at the time 

 of the survey; 



the standard deviation of the fluctuations in 

 logp c, owing to observation errors or changes 

 in the distribution of the stock; and 

 the relative vulnerability of each age class to 

 the fishery and the /"' survey, respectively. 



The corresponding negative logarithm of the sampling 

 density is 



