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



145 



sessment, and it may thus be unsuitable for more 

 general fisheries management problems. In order to 

 illustrate the potential pitfalls of the method, some 

 simulations were performed. The paper is presented 

 solely as a scientific appraisal of the suggested ap- 

 proach, and the examples given are purely illustra- 

 tive. We show how modifications of the technique can 

 lead to improved performance, but we do not formally 

 propose any alternative assessment methods here. 



The Monte Carlo approach 



Description 



Restrepo et al. ( 1992) described an approach to quan- 

 tifying the uncertainty in the results of sequential 

 population analyses for various fish stocks. They 

 motivated their approach by noting that 



Fisheries managers recognize the dangers of accept- 

 ing parameter estimates without consideration of the 

 variability inherent in the estimates offish stock status 

 and related parameters... If all sources of error are 

 not appropriately accounted for, then estimates of the 

 imcertainty in the assessment results may be too small. 



This Monte Carlo approach (hereafter MCA) proceeds 

 as follows. Probability distributions are used to de- 

 scribe uncertainty in the inputs to an assessment 

 model. These distributions are constructed in two ways: 



1 If observed data are available for a specific in- 

 put, a parametric statistical model for the data is 

 assumed, and the parameters are estimated by 

 maximum likelihood. A parametric bootstrap is 

 then used to obtain a sample of input values. 

 These values are used as values of the input in 

 the assessment model. 



2 If no data relevant to the input are available, a 

 subjective "prior" distribution, representing edu- 

 cated guesses about the true value, is placed upon 

 it.' A sample from this prior is used in the as- 

 sessment model. 



The second case occurs in many stock assessment 

 procedures. For instance, a prior was needed for natu- 

 ral mortality, M, in the swordfish assessment of 

 Restrepo et al. ( 1992). Similarly, we required a sub- 

 jective distribution for the growth rate parameter, 

 MSYR in the bowhead whale example in the section 

 "Application of MCA to bowhead whale assessment." 



The next step in MCA is to compare simulated 

 model outputs, such as a time trajectory of stock sizes 

 to observed data, in order to formulate a likelihood 

 (assuming lognormal deviations). Many input param- 

 eter sets are drawn randomly from the specified in- 

 put distributions (i.e. either from data-based boot- 

 strap or subjective prior), and for each set, a condi- 

 tional maximum likelihood estimate (MLE) is calcu- 

 lated for quantities of interest, given the fixed input 

 parameters and the observed data. The simulation 

 distribution of such conditional MLEs is used to quan- 

 tify uncertainty. The simulation is viewed as translat- 

 ing input uncertainties into output uncertainties.'^ 



MCA is suggested for situations where (possibly 

 many) nuisance parameters exist. These are typically 

 the model input parameters for which no informa- 

 tive data are available. The basic strategy is to esti- 

 mate the quantities of interest (e.g. current stock size 

 and production rate) conditional on values of the 

 nuisance parameters ajid then to integrate over the 

 prior for the nuisance parameters. The distribution 

 of the conditional estimates of the parameters of in- 

 terest is then examined for the purposes of inference. 

 For example, if 9y is an estimator of 6, conditional 

 on nuisance parameters y, then 



9a,=leyPiY)dY 



(1) 



is an MCA estimate of 9, where p(y) is the prior for y.^ 

 In practice, the integral in Equation 1 is not calcu- 

 lated exactly; it is approximated by using the Monte 

 Carlo simulation described above. In the form of an 

 algorithm, MCA proceeds as follows: 



1 Obtain the MLE 9^ from the observed data X and 

 a likely value Yq. 



2 Sample / from the prior p(7). 



3 Sample pseudo-data X* from a distribution with 

 density /"(a:; \jfiX)), where /"(x; i//) is a model for the 

 data but not necessarily the assessment model 

 and where i//(X) is an estimate of the parameters 

 of this model. i//(X) may depend on the results of 

 step 1, namely 9y and Yq, or even on standard 

 MLEs 6 and f- An example of /"would be to as- 

 sume X' ~ NiX, \j/), where y/ is an estimated dis- 

 persion matrix. 



' Such a distribution is effectively a Bayesian prior distribution. 

 However, since the framework of MCA is not Bayesian, Restrepo 

 et al. (1992) do not refer to such a distribution as a prior 



' Restrepo et al. (1992) also consider uncertainty in the assess- 

 ment model itself, but this issue is not of primary interest here. 

 In the bowhead whale application, the assessment model is fixed 

 by the IWC. 



■' In general these parameters may be multivariate vectors. For 

 simplicity, we restrict our focus to scalar parameters in the sub- 

 sequent examples. 



