Poole et al : A proposed stock assessment method and it application to bowhead whales 



149 



bowhead assessment, but it nevertheless captures 

 many of the essential features of the bowhead popu- 

 lation. The model is viewed as having two inputs (K 

 and MSYR) and one output (P1993); with fixed values 

 of MSYR and the initial K, Equation 3 is applied re- 

 cursively until P1993 is obtained. We use the term 

 "model input" for input parameters whose true value 

 is uncertain. Since the time series of catches C, is 

 exactly known, we regard it as a set of constants in 

 Equation 3, rather than as a set of model inputs. 



In our simplified version of the bowhead analysis, 

 we treated MSYR as a nuisance parameter ( yin our 

 previous notation) and K was the parameter of in- 

 terest (the 0from before). The only parameter about 

 which we observed data for a parametric bootstrap 

 was P1993. MSYR was assigned a subjective prior dis- 

 tribution. In terms of implementation, we ran the 

 model "backwards" with P1993 and MSYR as inputs 

 and K as output. A Newton-Raphson algorithm was 

 used to solve for K. As a result, we effectively had 

 P1993 and MSYR as model inputs, and K (obtained 

 conditionally on MSYR and the data) as the model 

 output. 



Implementation of MCA and a Bayesian approach 



To evaluate MCA, we examined its performance when 

 the true whale stock status was known. "True" val- 

 ues of K and MSYR were selected, and the PDM was 

 run to obtain the "true" value of P1993. Because K 

 (the parameter of interest here) has a "true" value, 

 we were able to assess the accuracy of the estimates 

 produced by the simulations. 



MCA was applied to the simple PDM of Equation 

 3 in a number of steps. The prior for the nuisance 

 parameter MSYR was gamma(8.2, 372.7), and the 

 likelihood for the observed total population in 1993 

 was A'^(Pi993,626-). These choices were based on IWC 

 consensus (IWC, 1995) and were the same as used 

 in previous work (Raftery et al.^ ). 



We assumed that we had a single observation from 

 the likelihood for P1993. In practice such an observa- 

 tion is usually obtained by means of a census. The 

 observation is typically a maximum likelihood esti- 

 mate of P1993, therefore we denoted it by P1993. 



An original conditional MLE was obtained by con- 

 ditioning on a "likely" point estimate of MSYR, say 

 MSYRq. We chose MSYRq = 0.02, the mean of the 

 prior for MSYR. The model was then run backwards 

 (i.e. with P1993 and MSYRq as inputs) and the likeli- 



hood maximized. The resulting output was the con- 

 ditional maximum likelihood estimate K j^syrq of K 

 because (given MSYRq) it lead to the value of P1993 

 that maximizes the likelihood. 



The MCA estimation then proceeds as follows: 



1 Draw P'i993 fromM Pi993,6262). This is the para- 

 metric bootstrap from a distribution with mean 

 given by the observed total population in 1993. 



2 Draw MSYR" from the prior for MSYR. 



3 Obtain /^msvr* by running the model backwards 

 with P'i993 and MSYR' as inputs. 



4 Repeat steps 1-3 many times to form a collection 

 of K j^fgYR* estimates. Like Punt and Butterworth 

 (1997), we used 1000 replications. 



5 Use Kjv/sy/} and the distribution of the K j^jgyn* 

 estimates to obtain inference about K. Specifically, 

 the distribution of the K i^syr* estimates shows 

 how the conditional MLE of K changes as MSYR 

 is varied according to its prior. 



For comparison with MCA, consider a fully Bayesian 

 analysis which involves specifying priors for every 

 model parameter, i.e. K, MSYR, and P1993. This in- 

 troduces an extra complication in that the input dis- 

 tributions and the model together induce a distribu- 

 tion on the output. There are thus two distributions 

 (the specified prior and the induced distribution) on 

 the output that need to be combined or reconciled in 

 some manner. For our "backwards" implementation 

 of the model here, the priors for MSYR and P1993 in- 

 duce a prior on the output K. This issue has received 

 considerable attention at the IWC and work in the 

 area is ongoing. A possible solution involving loga- 

 rithmic pooling of the two distributions is discussed 

 in Raftery et al.^ and Raftery and Poole. ^ 



For this example, it was useful to compare a Baye- 

 sian approach with MCA, but avoiding the added 

 complexity of the prior incoherence. This could be 

 achieved if we simplified the Bayesian analysis 

 slightly by ignoring the prior on the output K. We 

 had a prior for MSYR and a likelihood for P1993 as 

 we did in the MCA implementation above. In addi- 

 tion, we now also had a M 7800,13002) prior for Pi993. 

 This was the prior used in Raftery et al.^ and was 

 again based on IWC consensus. The prior and likeli- 

 hood were combined to 3deld a posterior distribution 

 for P1993. Because we had no data on MSYR, its prior 

 was not updated to a posterior. The only operational 

 difference between MCA and the Bayesian method 

 was in the generation of values for P1993: with MCA, 



•^ Raftery, A. E., D. Poole, and G. H. Givens. 1996. The Baye- 

 sian synthesis assessment method; resolving the Borel Para- 

 dox and comparing the backwards and forwards variants. 

 Paper SC/48/AS16 presented to the IWC Scientific Committee, 

 June 1996. 



Raftery, A. E., and D. Poole. 1997. Bayesian synthesis meth- 

 odology for bowhead whales. Paper SC/49/AS5 presented to 

 the IWC Scientific Committee, October 1997. 



