578 



Fishery Bulletin 105(4) 



The probability for all vessels (i.e., 

 written as 



strata) can then be 



P[X,=y„...,X,,=y„\T„...,T„ 



The parameters of the gamma prior, ^(A, I a, /3), can then 

 be estimated by maximizing the marginal likelihood 

 given above and arriving at MML estimates (or, /J). Initial 

 estimates for the gamma distribution can be provided 

 by moment estimators (Carlin and Louis, 2000) where: 

 r,=m;/T;, and r and s^ and the sample mean and vari- 

 ance of their, 1, and 



«„ 



-r^/(sf-ry\a/r-)/n)andPQ = r/d. 



The gamma prior is the conjugate distribution (Patrick, 

 1972; Carlin and Louis, 2000) for the Poisson sampling 

 distribution, which means that the posterior distribution 

 is in the same family as the prior distribution for each 

 stratum. The posterior distribution is p(AJ v,, of,/3, t,) = 

 fiy^, A^l d,P,ri)/g{yj\ q;,/3,t,), which can be verified to have 

 the gamma distribution g^(AJa'=v, + o,/J=l/(T, + l//5)). The 

 mean of this posterior distribution, providing estimators 

 for the A , can be calculated from 



A,=£(A;) 



: JA,p(A,ly„a,i3,r,)dA, 







= (y,+a)/(r,+l/^) 



or more simply can be recognized as the product of the 

 parameters of the posterior distribution. 



The conventional maximum likelihood (ML) estimator 

 of A; for the Poisson strata is ^,=y,-/T,. The EB estimator 

 of A, based on the mean of the posterior distribution 

 can be seen as the weighted average of the ML stratum 

 estimator and the mean of the gamma prior dji and will 

 lie between these two values. 



Simulation methods 



Simulation was performed on the Poisson-gamma EB 

 model described above. Each replication simulated the 

 seabird bycatch of 50 vessels, and was repeated 1000 

 times. Each replication assumed that the "true" bycatch 

 rate for each vessel (A,) was distributed as an observation 

 from the gamma distribution ^( A, I «= 0.603, /3=0.030); 

 whose parameters were estimated in the EB analysis 

 which follows. The number of hooks that were "observed," 



in thousands, was distributed uniformly as t/(0, T^a^), 

 with T^3^ = 1200, 500, 1000, 2000, 50001. For each of 

 these simulations then, t^^,^ = |100, 250, 500, 1000, 2500). 

 Finally, the number of "observed" seabirds iy^) was 

 simulated using the Poisson distribution with X-=kf€^, 

 where A- and t, were previously randomly generated as 

 described. 



For each replication, the simulated (>,, r,) were ana- 

 lyzed by using the empirical Bayes method, by first esti- 

 mating ( d,P) using the MML, and then using these pa- 

 rameters to calculate the EB estimate A^=(v,-i-ci:)/(T,+l/^). 

 The ML estimator for each stratum was A,=y,/T,, and 

 the global unstratified (GU) estimator was X=IyJlT-. 

 The performance of these estimators was measured by 

 using 



50 



TMSE- 



(X,-lf/50, 



where X^ could be any of A,, A^, or A. The simulation 

 was repeated 1000 times, and the TMSE values were 

 averaged to measure the overall performance of these 

 estimators. 



Analysis of bycatch data 



In 2002, The North Pacific Longline Association, which 

 has many longline vessel operators as members, vol- 

 untarily followed proposed regulations that required 

 the use of effective seabird avoidance gear during fish- 

 ing operations. These voluntary guidelines were imple- 

 mented into formal regulations in February 2004. 



EB analysis was performed on the bycatch of seabirds 

 from individual longline vessels fishing in the eastern 

 Bering Sea. The data were the annual observed by- 

 catches of seabirds (v, ) and the total number of observed 

 hooks in thousands (t, ) of individual fishing vessels for 

 2002 and 2003. The 2002 data were used to fit the EB 

 model, and resulting A, estimates were used to predict 

 the A, for 2003. As a comparison, a similar analysis 

 was performed on data collected from 1997 and 1998, 

 a time when many vessels did not use bird-avoidance 

 gear and when the bycatch rate of seabirds was much 

 higher than in 2002 and 2003. 



Results 



Simulation results 



All simulations consisted of 1000 replications as 

 described above. When measured by TMSE, the EB 

 estimator was clearly superior to both the maximum 

 likelihood (ML) and global unstratified (GU) estimators 

 (Table 1). This was true regardless of whatever value of 

 ''max w^^ used in the simulations. The ratio R=TMSE(?i.^)l 

 TMSEiA^) increased as t^^^^^ increased, but the values of 

 TMSE for GU remained constant. Note that the (dj) 

 appeared biased when t^^^ = 5000 (Table 1). 



