577 



Using the empirical Bayes method to estimate 

 and evaluate bycatch rates of seabirds 

 from individual fishing vessels 



Daniel K. Kimura 



E-mail address: dan.kimura@noaa.gov 



Alaska Fishenes Science Center 



National Manne Fistienes Service 



National Oceanic and Atmospheric Administration 



7600 Sand Point Way NE 



Seattle, Washington 98115-6349 



From the joint distribution defined 

 by h(y,e\ti)=f{y\e)g{e\t]), the mar- 

 ginal distribution of the observed v 

 can be derived by integrating out f): 

 m(y\ri)=jh(y,0\)])d0. The empirical 

 Bayes method arises from the recog- 

 nition that t] can be estimated from 

 wHyli)) by using the marginal maxi- 

 mum likelihood (MML) estimators 

 or related methods. Once fi is esti- 

 mated, the posterior distribution of 

 can be obtained by using the Bayes 

 rule, p{0\y,fi)=f{y\0}g{0\ rjl/wiyi n), 

 and an EB estimate of can be made 

 from this posterior distribution. 



Minimizing bycatch of seabirds is 

 a major goal of the U.S. National 

 Marine Fisheries Service. In Alaska 

 waters, the bycatch (i.e., inadvertent 

 catches) of seabirds has been an inci- 

 dental result of demersal groundfish 

 longline fishery operations. Notably, 

 the endangered short-tailed albatross 

 (Phoebastria alhatrus) has been taken 

 in this groundfish fishery. Bycatch 

 rates of seabirds from individual 

 vessels may be of particular interest 

 because vessels with high bycatch 

 rates may not be functioning effec- 

 tively with seabird avoidance gears, 

 and there may be a need for sugges- 

 tions on how to use these avoidance 

 gears more effectively. Therefore, 

 bycatch estimates are usually made 

 on an individual vessel basis and then 

 summed to obtain the total estimate 

 for the entire fleet. 



The empirical Bayes (EB) (Efron 

 and Morris, 1975; Casella, 1985) 

 method offers the possibility of im- 

 proving within-vessel bycatch es- 

 timates, with the assumption that 

 the individual vessel bycatch rate 

 of seabirds has a gamma prior dis- 

 tribution. With the resulting Pois- 

 son-gamma EB model, it is assumed 

 that each vessel's bycatch of seabirds 

 has a Poisson distribution condi- 

 tioned on the realized "true" bycatch 

 rate. The basic principle of the EB 

 method comes from the realization 

 that the parameters for the gamma 

 distribution can be estimated from 

 individual vessel bycatches, and that 

 the resulting EB estimators of indi- 



vidual vessel bycatch rates should 

 provide estimates of individual by- 

 catch rates that have smaller total 

 mean squared error {TMSE) than 

 the individual vessel bycatch rates 

 estimated independently. The inde- 

 pendently estimated individual vessel 

 bycatch rate is simply the bycatch 

 per thousand hooks fished for each 

 vessel. A more complete introduction 

 to the empirical Bayes method as it 

 has been applied to different types 

 of problems is provided by Ver Hoef 

 (1996). 



The goal of this note is to clearly 

 describe empirical Bayes estimation 

 and provide a detailed example of its 

 application to the problem of estimat- 

 ing seabird bycatch. It is to be hoped 

 that a better understanding of the 

 theory underlying empirical Bayes 

 methods will lead to more applica- 

 tions in the area of fisheries man- 

 agement. 



Materials and methods 



General theory 



Mathematically, the empirical Bayes 

 (EB) method can be described as a 

 statistical procedure that has clearly 

 defined steps (Carlin and Louis, 

 2000). Let the prior distribution 

 of a parameter 6 (the parameter of 

 greatest interest) be g(0li)), where 

 the I) are unknown parameters, and 

 the sampling distribution for each 

 stratum observation y is fiy\0). 



The Poisson-gamma empirical 

 Bayes model 



The Poisson-gamma model is ideal 

 for illustrating how to calculate EB 

 estimators from the general theory 

 because the all the required integrals 

 result in a gamma function. For this 

 model, denote the gamma prior for 

 the seabird bycatch rate of vessel i as 

 g(X^\a,li), and the Poisson sampling 

 distribution as f(y^\k^,T^), where y, is 

 the number of seabirds observed, and 

 T; are the number of hooks observed. 

 The joint distribution of y, and A, is 

 then 



1 , „ exp(-A,rXA,r )^' 



y,'- 



A°-^exp(-A, //J) 



fora,/5,A, >0,y, >0. 



The marginal distribution is calcu- 

 lated by integrating out \: 



m(y^ Ia,i3,r, ) = 



y^T(a)p'' (x^+lipf^* 



Manuscript submitted 2 April 2007 

 to the Scientific Editor's Office. 



Manuscript approved for publication 

 16 May 2007 by the Scientific Editor. 



Fish. Bull. 105:577-581 (2007). 



