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 incidental result of demersal groundfish longline fishery operations. Notably, the endangered short-tailed albatross (Phoebastria albatrus) 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 effectively with seabird avoidance gears, and there may be a need for suggestions 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 improving within-vessel bycatch estimates, with the assumption that the individual vessel bycatch rate of seabirds has a gamma prior distribution. With the resulting Poisson-gamma EB model, it is assumed that each vessel's bycatch of seabirds has a Poisson distribution conditioned 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 individual vessel bycatch rates should provide estimates of individual bycatch rates that have smaller total mean squared error (TMSE) than the individual vessel bycatch rates estimated independently. The independently 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 estimating seabird bycatch. It is to be hoped that a better understanding of the theory underlying empirical Bayes methods will lead to more applications in the area of fisheries management.

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 [theta] (the parameter of greatest interest) be g ([theta]|[eta]), where the [eta] are unknown parameters, and the sampling distribution for each stratum observation y is f(y|[theta]). From the joint distribution defined by h(y, [theta]|[eta])= f(y|[theta])g([theta]|[eta]), the marginal distribution of the observed y can be derived by integrating out [theta]: m(y|[eta])=[integral]h(y, [theta]|[eta]) d[theta]. The empirical Bayes method arises from the recognition that [eta] can be estimated from m(y|[eta]) by using the marginal maximum likelihood (MML) estimators or related methods. Once [??] is estimated, the posterior distribution of [theta] can be obtained by using the Bayes rule, p([theta]|y, [??])=f(y|[theta])g([theta]|[??])/m(y|[??]), and an EB estimate of [theta] can be made from this posterior distribution.

The Poisson-gamma empirical Bayes model

The Poisson-gamma model is ideal for illustrating how to calculate EB estimators from the general theory because...