LORD: OPTIMUM DATA ACQUISITION 



lined. Consider the start of the Ath time period 

 where 1 ^ A' ^ m but otherwise arbitrary. The 

 management biologist has at his disposal data 

 observed through the (k - l)st period which will 



be designated by (Fq' Xi'- -Zfe-i)- '^^^^ ^^ ^" 

 abstract designation for data which may typi- 

 cally be in the form of catch reports, catch per 

 unit effort data, tower counts, etc. By conven- 

 tion Yq represents the preseason information, 

 such as the high-seas forecast, that is obtained 

 prior to the start of the run. However, all future 

 outcomes must be considered since the loss 

 function for the escapement is formulated in 

 terms of the final state of the system. The vector 

 £ , which characterizes both the run and the 

 corresponding set of optimum allocation rules, 

 is, from the biologists' point of view, an un- 

 known parameter whose value he is attempting 

 to infer. Generally may be considered to 

 have some underlying prior distribution which 

 may be inferred from historical data, etc. As 

 additional data are gathered, the probability 

 density of 6 may be successively updated to 

 reflect this additional information. Thus the 

 probability density for 0^ at the beginning of 

 the kth period may be written as 



f,,. (0 



Xo^ 



Y,....Y,_,-<) 



where the prior data, (Yq, Y^, ... y,,_j ), and the 

 manner in which it is obtained, (;, appear as 

 conditioning quantities. 



Now consider the distribution of the actual 

 allocation i]-. Generally Vj will be a random 

 variable whose distribution will depend on the 

 action taken, 5-, and on the true state of nature, 

 £. Thus the probability density of 77. may be 

 written in conditional form as g,(^,IS,>£)- 

 This tacitly assumes that the allocation result- 

 ing from a decision taken during any particular 

 time period is independent of the outcomes dur- 

 ing any other time i)eriod which in turn implies 

 that the individual fish is vulnerable during only 

 a single time period. This condition is generally 

 fairly well satisfied in most of Bristol Bay 

 where the fishing districts are relatively small 

 and the fish do not delay in their ujistream 

 migrations. Exceptions occur occasionally dur- 

 ing extreme tides when the fish may enter and 

 leave the fishery more than once before pro- 

 ceeding upstream. A similar exception would 



occur if a fishing district were of sufficient size 

 that individual fish must necessarily spend 

 more than a single time period in it. In these 

 more general cases we must include all prior 

 allocations as conditioning quanitities, i.e., the 

 a])i)ropriate density would be of the form 



g/(T?,|T?i,T?2v'i7/_i,5,-,£). An equivalent 



but more concise notation would be to condi- 

 tion the distribution of 77- by 5. and by the 

 state of the system at the start of the /th time 

 period, S-, i.e., g- {rj-\8-,S-). That this is an 

 equivalent conditioning follows from our pre- 

 vious definition of the state of the .system as 

 reflecting the true state of nature as well as the 

 effects of all previous policy decisions. If we 

 retain the assumption of independence of the 

 allocations, the joint probability density of 

 (17., 772,. -T?,,,) may be written in factored form 

 as 



m 



g('?i,i?2.-'?ml5i.52. .5„;e)»77g,(')il«,,9). 



(5) 



It is now possible to construct the risk func- 

 tions appropriate for the start of the kth time 

 period where, as usual, k is arbitrary. The ex- 

 perimental cost, C( ^ ), has been assumed fixed 

 in advance in which case it is equal to its ex- 

 pected value. Thus the first term of the risk, 

 jRj ,j ( O, is given simply by 



^i.fe (0 = ceo 



(6) 



for all k. 



The risk function associated with the catch 

 may be thought of as consisting of two parts. 

 The first part is the risk corresponding to the 

 loss already accrued through the kth time pe- 

 riod for which the management decisions have 

 already been made. The second is the risk over 

 the remainder of the run for which the decisions 



^k + i^ ^k + 2'--^m I'eiT'ai'i to be made. From 

 (2) and (5) we obtain 



S^J^-^^'^^-^'^O'Ii'-Ife-r^) 



1 = 1 



Jc^T?, (17,(0)- 77,) n, (0)g,(77, I5,,£) 



(7) 



1033 



