LORD: DECISION THEORY APPLIED TO SALMON FISHERY 



as the posterior density for A^ at the start of the 

 kth fishing period. The parameters are given by 



Ok - 1 = ao + iji + y2 +   • +yk - 1 and fik -i = 

 fio + e(p2 + P2 + . . . + Pit-i)- At this point it is 

 appropriate to observe that, as time progresses 

 and additional population data are obtained, the 

 distribution of A^, as specified by the parameters 

 afc_i and fik-i> will more and more reflect the 

 in-season sampling data with a corresponding 

 decrease in the relevance of the preseason infor- 

 mation implied by a^ and Pq. 



The probability densities given by Equations 

 (7), (10), and (14) enjoy a peculiar relationship in 

 which the posterior density of A'^, given by Equa- 

 tion (14), is from the same family as the prior 

 density, Equation (7), for the particular sampling 

 distribution given by Equation (10). Such pairs of 

 densities are called conjugate pairs (DeGroot 

 1970:159-166). It is obvious that one cannot, in 

 general, be so fortunate as to have parameter and 

 sampling distributions that form a conjugate pair 

 as in the model assumed' here. However, DeGroot 

 does outline some somewhat ad hoc procedures for 

 constructing reasonable posterior probability 

 densities. 



All of the quantities and distributions necessary 

 to compute the average or expected loss, i.e., the 

 Bayes risk, are now available. The expression for 

 the Bayes risk, to be evaluated at the start of the 

 A:th fishing period, may be written formally as 



Rki^u ^2, •••5m 1^0,2/1. ■■■yk -l) 



= J^fkiN\yo,yu  • •yk-i)dH/^dri, . . 



1 m 



Jd-nmUEJ n ^.(^. 1^) 



(15) 



i = 1 



where Qi , L(£^), and^. are given by Equations (9), 

 (6), and (14) respectively. Notice that the Bayes 

 risk as given by Equation (15) is a function not 

 only of the decisions already made, 6i, 82, ... 

 6;, _ 1 and the decision about to be made, 6;^ , but of 

 all future decisions 6^+1, ... 8^ as well. This 

 dependence on all decisions, past, present, and 

 future, reflects the assumption that the loss is a 

 function primarily of the final state of the system, 

 i.e., to a first approximation one cannot ascribe 

 values to individual units of escapement during 

 the season but only to the final total escapement. 

 This presents no particular analytical difficulties 

 since any particular sequence of optimum future 

 decisions S^ +1, . . . 5„ is certainly subject to revi- 



sion as time passes and additional information 

 becomes available. 



Substituting Equations (6), (9), and (14) in 

 Equation (15) gives 



Rk{8i,82, ...8„ 12/0,2/1, •• •2/fc-i) 



L- (Eo.E,. E..„)-iali,e-i"<-E.) 



m 



^ = ' r(.jr(/i,) 



■n; 



«", - 1 



(l-7j)«ri 



(16) 



where L'{Eq, E„ _i, E,^_2,    K-k) denotes that 

 portion of the loss function that does not depend 

 on E„. Thus L' is a fixed quantity and may be 

 removed from the integral signs. This leaves only 

 probability densities, which must integrate out to 

 unity, so that 



Rki8i,82,..-8m\ 2/0,2/1,... 2/k-l) 



(/Sfc_i)^-. 



1 («fc - 1) 



m 



where the escapement £„ has been expressed as 



m 



En = N^ pfl, . 

 i = 1 



The integrations in Equation (17) cannot be 

 performed as expressed. If the order of the inte- 

 grations is reversed, the integration with respect 

 to A^ may be performed but the remaining in- 

 tegrations over fji, 7J2, ... t)„ will be virtually 

 impossible. However, if the exponential term 



exp (-6A^2 P,^,) is expanded in its Maclaurin 



series and if the resulting multinomials of the 



/ III \n 



form -^I-bN^ p,Vij {n = 0, 1, . . .) are ex- 

 panded according to the multinomial theorem, the 

 integrand in Equation (17) will be in a completely 

 factored form. As a result of this factorization, the 

 integrals take the form of various moments about 



841 



