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Fishery Bulletin 90|4). 1992 



In practice, however, the time and computer re- 

 sources required to carry out such a large-scale simula- 

 tion make it more practical to derive the input uncer- 

 tainty distributions from parametric statistical analyses 

 of data. This would involve assuming a distribution type 

 for the inputs and estimating their mean and variance. 

 For example, by virtue of the central limit theorem, 

 an estimated mean has an approximately normal 

 distribution if the sample size is sufficiently large. 



Often, the distributions determined for some of the 

 inputs will not be based on a rigorous statistical treat- 

 ment of the data, but rather will represent educated 

 guesses about the likelihood of the inputs taking on par- 

 ticular values (this is probably most true for the natural 

 mortality rate, M, which is usually assumed and not 

 estimated). The outputs would then represent the ana- 

 lyst's personal uncertainty in the assessment results. 



The above approach can be generalized to allow for 

 uncertainty in the formulation of the SPA model as 

 well. Suppose one believes that there is a 70% chance 

 that the fishing mortality rate in the last year does not 

 decline with age after a fully recruited age (this is often 

 known as a "flat-topped" partial recruitment curve), 

 and a 30% chance that it does decline ("dome-shaped" 

 partial recruitment). Then one could conduct 70% of 

 the simulations with an SPA that assumes the flat- 

 topped curve and 30% with the dome-shaped curve. 



The resulting combination of outputs would reflect the 

 intuitive estimate of uncertainty about the SPA model 

 formulation. Similarly, the approach can also account 

 for uncertainty concerning data-dependent decision 

 making. For example, if several abundance indices are 

 available, one might subject each index to a preliminary 

 test to decide whether the index is acceptable for 

 calibrating the SPA, e.g., via analysis of residuals. One 

 can repeat this decision-making process for each of the 

 simulated data sets and thus account for the uncertain- 

 ty associated with screening indices. 



In summary, the Monte Carlo approach to quantify- 

 ing uncertainty consists of generating a large number 

 of pseudo-data sets, drawn at random from specified 

 distributions, and carrying out the entire assessment 

 procedure for each data set. The distributions of the 

 assessment outputs and derived statistics are then simi- 

 marized, e.g., as histograms. The simulation is thus 

 viewed as a means for translating input and model 

 uncertainties (measured and/or perceived) into output 

 uncertainties. 



Analyzing the consequences 

 of management options 



Estimating risl<s 



The simulation results can also be used to quantify the 

 uncertainty associated with a future management ac- 

 tion. An example is the determination of uncertainty 

 in the catch of the current year that would maintain 

 the fishing mortality at the level of the previous year 

 (Fstatus quo)- A point estimate might be computed as 

 follows. An SPA of some sort is used to estimate the 

 population size at the end of the previous year and the 

 fishing mortality during that year. Assuming that 

 recruitment in the current year is equal to the long- 

 term average, the estimated population size in the cur- 

 rent year can be computed. Finally, the harvest which 

 causes the population to experience the same fishing 

 mortality rate as that estimated for the previous year 

 can be computed. To quantify the uncertainty in this 

 result, the whole procedure can be repeated 1000 times, 

 each time perturbing each input to the SPA by a ran- 

 dom amount (as per the specified uncertainty distribu- 

 tions). This results in 1000 sets of estimates of popula- 

 tion size, fishing mortality, and natural mortality rate 

 which, together with a set of randomly-drawn values 

 for recruitment, can be used to generate 1000 esti- 

 mates of the total allowable catch which will cause the 

 fishing mortality to remain unchanged. These values 

 can be organized into a relative-frequency histogram 

 such as in Figure 1. 



From the histogram, it appears that the most likely 

 (modal) value for achieving the management goal 



