Thompson Bayesian approach to management advice from a simple dynamic pool model 567 



1 1 



/ B(Fmsy) \ /l-2v'\i-'" / B(Fmsy) \ I (l + m)(l-2v') \^"^ 



lim — ^ ^^^' = and lim — ^ = \— ~ ^ . (27) and (28) 



K--0 \B(FMELsy)/ \ 1-v' / K-oc \B(Fmelsy)/ \(l + m)(l-v') - v'/ 



Equations (26-28) decline from a value of 1 at v' = to a minimum at v' = v'o. The minimum value depends on 

 K" and m, but is never greater than 1/e. 



Estimating the parameters of the beta distribution 



To fit the Gushing stock-recruitment curve to a set of n stock-recruitment data points, it seems reasonable to 

 assume the following model: 



Yi = P -H qxi -I- £i, (29) 



where x; represents the natural logarithm of the ith stock biomass datum, y; represents the natural logarithm 

 of the ith recruitment datum (lagged according to the age of recruitment), p = ln(p), and £, is an independent error 

 term distributed as N(0,o-). 



Press (1989) presented a Bayesian approach to estimating the parameters of the pdf of q using Equation (29) 

 as the underlying model. The following paragraphs summarize this presentation, which begins by rephrasing the 

 problem in the form of Bayes' theorem: 



h(q,p,o I X, y) oc 



|~| f(yi I Xi,q, p,o) 



i = l 



gi(q) g2(p) g3(o), (30) 



where x is the vector (xj, . . ., Xn)'; y is the vector (yi, . . ., yn)'; h(q, p,o | x, y) represents the posterior pdf 

 of the parameters q, p, and o; f(yi | Xi,q, p,o) represents the conditional pdf of yj given the observed value of X; 

 and any particular values of q, p, and o; and gj() represents the prior pdf of the jth parameter. 

 Given the assumptions implicit in Equation (29), f(yi | Xi,q,p,a) can be written 



/- (yi-p-qx,)-\ 



exp 



2o2 

 f(yi I x,,q,p,o) = ^ —^ '-. (31) 



A special case of interest is the one in which the gj(-) are all "vague" (also called noninformative or indifference) 

 priors. These are pdfs which reflect indifference regarding the probability of alternative parameter values. Press 

 (1989) treated gi(q) and g2(p) as constants, implying that all values on the real line are equally likely in the prior 

 distribution. Since a is constrained to be positive, however. Press set g3(o) = l/o, reflecting a uniform prior 

 distribution for ln(o). 



Using Equation (31) and the priors specified by Press (1989), Eq. (30) gives a straightforward solution. The 

 classical least-squares estimates of q and p (q and p, respectively) obtain as the maximum-likelihood estimates. 

 In their posterior pdf, q and p jointly follow a bivariate Student's t distribution, so that marginally the posterior 

 pdf of q, hi(q | x, y), follows a univariate 3-parameter t distribution with n-2 degrees of freedom: 



n-l\ 

 hi(q I x,y) = , (32) 



n(n- 2) si 1 -i- 



M (n-2)s2 



