LORD: OPTIMUM DATA ACQUISITION 



t, is usually fixed prior to the start of fishing 

 and, except for minor variations, remains essen- 

 tially unchanged during the run. We define the 

 optimum experimental design, ^ *, by 



min/?Q (6i,§2'-^m'^) " ^o i^i^^2^-^m'^^ ) 

 ^ (16) 



where the overbars on the 'l ^, } denote aver- 

 aging overall allowable decision rules 6 in the 

 set D and the risks are determined prior to the 

 taking of any observations. Similarly, the op- 

 timum decision rules (5^, Sg, •••S*j) are 

 that set of decisions that minimizes the average 

 risk over the duration of the run. 



DISCUSSION 



A mathematical description of a salmon fish- 

 ery that includes both stochastic and dynamic 

 elements has been formulated although the final 

 result is relatively general and somewhat ab- 

 stract. Indeed, the mathematics was formulat- 

 ed specifically to simulate the actual assess- 

 ment and management of the typical Bristol 

 Bay sockeye salmon fishery. The statistician or 

 management biologist periodically acquires ad- 

 ditional data, such as catch reports and test 

 fishing results, from which he can make repeat- 

 edly more refined estimates of the true state of 

 nature. Also, although perhaps quite uncon- 

 sciously, he attempts to estimate the losses (or, 

 if an optimist, the gains) associated with any 

 course of action and the relative probability of 

 occurrence of these losses. Then, based on all 

 data obtained to date, including all past deci- 

 sions and outcomes, he attempts to formulate 

 a future strategy that will minimize his risk. 

 The analysis of the preceding section attempted 

 to express this sequence of events in a more 

 formalized and quantitative manner. 



The apparent fidelity of statistical decision 

 theory to the real world suggests that it pro- 

 vides a very general theoretical tool for the 

 description of such processes. However, the im- 

 plementation of such a theory may give rise to 

 some practical problems of considerable difficul- 

 ty, some of which were discussed in the Intro- 

 duction. In particular it was emphasized there 

 that the ability to formulate a problem as a 



dynamic program or, almost equivalently, as a 

 problem in sequential statistical decision theory, 

 by no means assures that a solution may be 

 obtained. In this section some additional gen- 

 eral features of dynamic programming, as they 

 apply to the fisheries management problem just 

 formulated, will be further elaborated. 



The set of optimum decision rules has been 

 defined as that set that minimizes the Bayes 

 risk over the duration of the run. From this it 

 follows that the kth decision must be chosen 

 optimally as a function of the set of prior ob- 

 servations (^O'Zi'- Zfe-i) ^"^ ^^ ^ function of 

 all prior decisions (5^, 62,- -S^.i)- In other 

 words, 6 must be chosen at each stage in an 

 optimal manner taking into account all prior 

 observations and decisions. At this point we 

 continue Hillier and Lieberman's (1967) char- 

 acterization of dynamic programming, the first 

 three principles of which were presented in the 

 Introduction. Their principle number four states 

 that: Given the current state of the system, an 

 optimal policy for the remaining stages is inde- 

 pendent of the policies adopted in the previous 

 stages. This is a paraphrase of the fundamental 

 "Principle of Optimality" of Bellman (1957, 

 p. 83) which states that: "An optimal policy 

 has the property that whatever the initial state 

 and initial decision are, the remaining decisions 

 must constitute an optimal policy with regard 

 to the state resulting from the first decision." 

 The principle of optimality thus assures that 

 the policy we have specified is indeed an optimal 

 policy. 



The contradiction, which is more apparent 

 than real, will now be resolved between the 

 principle of optimality just stated and our 

 previous contention that the choice of an optimal 

 decision 5,^ depends not only on (yQiXi'-- 

 F^_i)but also on the previous decisions ( 6^, 

 ^2'--^fe-i)- Recall that the state of the system 

 at time k, say, S^, is assumed to be uniquely 

 determined by (Yq, Yi,...Y,,.i), (5^,52,- 

 5^_j), and , the true state of nature. The 

 converse is not true, however, since a multi- 

 plicity of different decisions and observations 

 may lead to the same S^, i.e., there is generally' 

 no unique path to S^. Thus, while it is perhaps 

 more appropriate to state that the optimum 

 Sfe is a function only of S^, it should be borne 



1035 



