FISHERY BULLETIN: VOL. 74, NO. 4 



the origin. These integrals are all standard forms 

 (c.f., Bierens de Haan 1939). The reader will be 

 spared the details of this reduction and the ensu- 

 ing integrations. The final expression for the 

 Bayes risk is 



Rk (5i,52 5„ \yQ,y^ Vk -\) 



= L'iE^,E,_„...E„-K) + 



tti 



■k -1 



Pi"; 



a 



fik - 1 j^i J'i + HAj 

 . n 



b \ r(afc _ 1 + n - 1) 



fik - 1 n=o \ ^k-ij n\ r(afc _ i) 



2fc, = n 

 m 



m 

 \k1IC2   . fCfn/i = 1 



'i + th + ki 



fr r{v, +11,) n., -hA^) ^k, 



r=\ r(.,)r(., +11, + k,) ' 



(18) 



n 



where ( kik2 ... k„,) denotes a multinomial 

 coefl[icient. 



A slightly different form for the risk may be 

 obtained under an alternate set of assumptions. 

 Considerable emphasis has heretofore been placed 

 on the conjugacy of the gamma-Poisson families 

 of distributions. The gamma-Poisson assumption 

 is a reasonable one and the resulting conjugacy 

 lends a certain elegance. However, this line of 

 analysis results in posterior gamma parameters 

 («fc, Pk) that, among other things, depend on the 

 run fractions (ft) {i = 1, . . . k). This parameter 

 dependence on the run fractions virtually pre- 

 cludes treating the set (ft) (i = 1, . . . k) as any- 

 thing but fixed quantities; i.e., once a variable 

 becomes the argument of a gamma function one 

 has usually arrived at an analytical dead end. In 

 actual practice, however, the quantities (p; ) (i = 1, 

 . . . m.) are random variables since there may be 

 considerable year-to-year variation in the time 

 profile of the run. Such temporal variation may be 

 of considerable importance in Bristol Bay because 

 of the large magnitude of the run and its short 

 duration. 



It has been suggested (0. A. Mathisen, pers. 

 commun. and others) that the probability density 

 of A^ is most appropriately conditional upon the 

 catch-per-unit-effort (CPUE) observed during the 

 course of the run. In so doing one can remove the 

 explicit dependence of (a^ , P^ ) on (ft ) {i = 1, . . . k). 

 An implicit dependence remains, however, since 

 the CPUE will be a function of the run fractions. 

 One can formally bypass this dependence, how- 

 ever, by relating the density of A^ directly to the 



842 



CPUE. In so doing one can then introduce tempo- 

 ral variability in the set (ft) (i = 1, . . . m) and in 

 evaluating the Bayes risk an additional expecta- 

 tion with respect to the density of these random 

 variables must be taken. 



An almost ideal probability density to describe 

 the run fractions is the Dirichlet density defined 



by 



,, . r(Yi + Y2 + ... + Ym) 



/l(Pl,P2, . ..pm) = 



r(Yi)r(Y2) . . . r(Y„; 



ip ^. 



Pm 



y - 1 



(19) 



where ft^O for all i. As written this density is 

 singular since the variates must satisfy the side 



condition V Pj = 1. The choice of the pa- 



1 = 1 

 rameters (yi, Y2. .•• Ym ) then permits the 

 specification of any m of the means, variances, and 

 covariances of the (ft) (i = 1, . . . m). If Equation 

 (19) is substituted in Equation (16) the integra- 

 tions with respect to A'' and (tji, tjo, . . . tj,„ ) may be 

 done as before. The remaining integrals over (pi, 

 P2) ••• Pm) are all Dirichlet integrals (Wilks 

 1962:177, et seq.) for which the values are readily 

 determined. The resulting Bayes risk for this case 

 may then be shown to be given by 



/?,(6,.5o. ...5„, ICPUE) 



= L'iEo-E,, _ 1. . . . E„ _a) + -^ V — '-^— 



a ^ / b Y r(a, _, + n + l 

 k-o\kik2.. ./c,J 



fj r(f, +^•)^(., +iu,)r(Y, + k) 



1) 



i = 1 



r(.,)r(., -Hrt +^)r(Y,) 



y (Y, + k,){p, + k,) 



f, + Ph + k, 



) = 1 



(20) 



where G = X 



Yi 



i = 1 



Equations (18) and (20) are somewhat in- 

 timidating, particularly if one were to attempt to 

 infer the qualitative behavior of the system as the 

 parameters descriptive of the fishery and its 

 management are varied. Indeed, Equations (18) 

 and (20) are virtually useless for this purpose with 

 the exception of the determination of certain 



