Thompson: Bayesian approach to management advice from a simple dynamic pool model 563 



To incorporate the logarithmic loss concept into the model, first note that Equation (1) allows z(F, q) to be written 



1 



r/D\ /i+K"+F'\ii-<5 



F 

 Y(F) L\M/\ (1-F')^ /J F' /1 + F'msy\M 1 + K" + F' \i-q 

 z(F, q) = ^-^ = —^ = 3 I 1^ I r-zr. 1 • (5) 



MSY 



MSY 



fp \/ 1 + K" + F'msy \ 

 ,M (1 + F'msy)^ 



j_ F' 



l-q 



MSY 



1 + F' 



1 + K" + F' 



MSY, 



For an arbitrary value of q, the (logarithmic) loss associated with a given choice of F is thus 



,,,^ ,, , ,!.' ^ 2 1n(l + FMSY) - ln(l + K" + FMSY) ,,„,, 2 1n(l + F') - ln(l + K" + F-) 



L[z(F, q)] = In(FMSY) ln(F ) + : • (6) 



l-q 



l-q 



Substituting Equation (6) into Equation (4), the risk can be written 



i.,Tr.T. M, r'i>. ^(w^>' ^ 2 1n(l + F'MSY) - ln(l + K" + F-msy) \ , 

 E{L[z(F, q)]} = I P(q) ln(F msy) - dq 



Jo l-q / 



/, 



P(q) In(F') 



2 ln(l + F') - ln(l + K" + F')\ 

 l-q 



dq. 



(7) 



From Equation (2), it is clear that F'msy involves only K" and q. Thus, regardless of the form of P(q), the first 

 integral on the right-hand side of Equation (7) is independent of F. Therefore, the problem of finding the Bayes 

 decision is equivalent to minimizing the second integral on the right-hand side of Equation (7). Remembering that 



the integral (taken over the interval to 1) of a cons- 

 tant multiplied by P(q) is equal to the constant itself, 

 the following proxy objective function is obtained: 



Llz(F,q)l 



logarithmic 



Figure 1 



Three possible loss functions. Loss, or relative utility foregone, 

 is plotted against the ratio of Y(F)/MSY for quadratic, linear, 

 and logarithmic loss functions. 



Ei{L[z(F, q)]} = -ln(F') + 



' P(q) 



[21n(l-t-F') - ln(l-hK"-HF')] I dq. (8) 



J q 



Incorporating a beta 

 probability density function 



The next step in determining the Bayes decision is to 

 select a form for the pdf P(q). Bayesian decision theory 

 frequently makes use of the beta family of pdfs (e.g., 

 DeGroot 1970, Holloway 1979). The beta distribution 

 would seem to be a natural candidate for P(q), since 

 it constrains q to the necessary (0,1) range. In its 

 standard form, the beta distribution can be written 



