The IUCN/SSC Shark Specialist Group
Shark News 13: July 2001
|
Bayesian methods in shark fishery management
Elizabeth A. Babcock and Ellen K. Pikitch,
Marine Conservation Programs, Wildlife Conservation Society, USA.
Introduction
Compared to classical statistics, Bayesian statistics are a fundamentally different way of approaching parameter estimation, model fitting, and hypothesis testing. In the last decade, Bayesian statistics have gone from an obscure method advocated by a minority of statisticians, to a standard method in fisheries stock assessment. There are two main reasons for the increasing popularity of Bayesian statistics.
- Bayesian methods allow the inclusion of information from diverse sources through the use of prior probabilities.
- Bayesian methods provide results in terms of probability distributions, which can be used in the decision analyses that assessments must supply for fisheries management.
These advantages are particularly relevant to shark fisheries, where data are generally poor, and many stocks are badly depleted. To demonstrate these points, we will describe the Bayesian surplus production model for large coastal sharks in the U.S. Atlantic and Gulf of Mexico from the 1998 Shark Evaluation Workshop (NMFS 1998).
Surplus production models in the large coastal shark fishery
The U.S. Atlantic and Gulf of Mexico large coastal shark fishery is dominated by sandbar Carcharhinus plumbeus and blacktip Carcharhinus limbatus sharks. The fishery is managed by a quota on commercial shark landings, and bag limits in the recreational fishery. Previous assessments of large coastal sharks (NMFS 1998, 1996) used a simple surplus production model.
Surplus production models are based on the following equation (paraphrased from Hilborn and Walters 1992).
Population next year = population this year + surplus production - catch
Assuming logistic population growth, the population's per capita growth rate will be highest at low population levels, approaching the intrinsic rate of growth (r). However, the total surplus production is highest when the population is at half of the carrying capacity (K). At K/2, the surplus production is rK/4, so the maximum sustainable yield (or catch) is also rK/4.
The data required to estimate the parameters of this model are the total catch in each year, and at least one time series of estimates of relative abundance, such as a catch per unit effort (CPUE) series.
A standard method of estimating r and K is to assume a population level at the beginning of the CPUE time series and use the logistic model to predict the whole time series, with an assumed value for r and for K. The best estimates of r and K are those that cause the predicted time series and the observed time series of CPUE to be the most similar (Hilborn and Walters 1992). There are several ways of finding these best fit values, including minimizing the sum of the squared differences between the observed and predicted CPUE values (sum of squares estimation), and finding the estimates that maximize the likelihood of having observed the data given the parameters (maximum likelihood estimation).
This method works well if the data are informative, meaning that only a small range of parameter values provide a good fit between the observed data and the predictions of the model. Unfortunately, for large coastal sharks in the U.S Atlantic and Gulf of Mexico, the catch per effort data are not informative with respect to the critical parameters, r and K. The population began in the 1970s at an unknown level and declined through the 1980s and 1990s. This so-called "one-way-trip trajectory" is a textbook example of uninformative data (Hilborn and Walters 1992). These CPUE data could be from a relatively unproductive population with a high starting biomass (low r and high K), or from a productive population with a low starting biomass (high r and low K). A joint likelihood profile of r and K (Figure 1) shows this. The darker regions in the graph represent combinations of r and K that provide a better fit between the model and the data.
Figure 1. Joint likelihood profile of r and K
for combined large coastal sharks, from
McAllister and Pikitch 1998a.
|
In the 1996 assessment of large coastal sharks (NMFS 1996) the surplus production model's maximum likelihood estimate of r was 0.26. However, demographic analyses of sandbar and blacktip sharks presented at the same meeting indicated that r was probably less than 0.10, based on age-at-maturity, litter size, and other life history characteristics. The discrepancy between the demographic and surplus production estimates of r are easily explained by the uninformative CPUE data-while an r of 0.26 was the best fit estimate, the fit for r=0.1 or even r=0.05 was not significantly worse (Figure 1). Clearly, the demographic information about r should be incorporated into the stock production analysis, and this is precisely what a Bayesian prior will allow.
Bayesian statistics
Bayesian statistics are not just another statistical model; they represent a fundamentally different approach to parameter estimation (Dennis 1996). Classical or "frequentist" statistics consider a parameter, such as r in the surplus production model, to be an unknown constant, while the data are considered to be realizations of a random variable. Frequentist statistics can calculate the probability of a certain set of data being collected given a certain set of parameters, but cannot assign probabilities to parameter values (McAllister and Kirkwood 1998). Thus, a classical 95% confidence interval (CI) does not imply that there is a 95% chance that the interval contains the true parameter value. Rather, it implies that, if data were collected and the analysis performed many times, 95% of the calculated CI's would contain the actual parameter value. Bayesian statistics, on the other hand, consider a parameter to be a random variable with a distribution that reflects the uncertainty about the parameter (McAllister and Kirkwood 1998). So, unlike a frequentist confidence interval, a Bayesian 95% CI can be interpreted as having a 95% chance of containing the true parameter value. For more information on the theoretical differences between Bayesian and frequentist statistics, see McAllister and Kirkwood (1998) and Punt and Hilborn (1997) for the Bayesian perspective; see Dennis (1996) for the frequentist perspective.
Bayesian estimation calculates a joint probability density function (pdf) of the parameters given the data, called the posterior distribution because it is the pdf after the analysis. The pdf of the parameters before the analysis is called the prior distribution. The posterior is calculated with Bayes' rule, which states that:
The posterior probability of the parameters is proportional to the likelihood of the data times the prior probability of the parameters.
The likelihood of the data given the parameters is a probability density function that provides a measure of the fit between the data and the model, given an assumed set of parameters. More formally, this likelihood function represents the probability of obtaining the observed data if the assumed set of parameters happened to be the true ones. Thus, the posterior pdf of the parameters is a function of both the fit between the data and the model, and the prior information about the parameters. If the data are very informative, then the posterior pdf will be determined by the data, and the prior will have little effect. Conversely, if the data are uninformative, the posterior pdf will resemble the prior (McAllister and Kirkwood 1998). In some cases, the posterior can be calculated analytically, however, for most fisheries models, the posterior must be calculated with a numerical integration method, such as the Markov Chain Monte Carlo (MCMC) or the Sampling Importance Resampling (SIR) algorithm (McAllister and Kirkwood 1998).
Much of the work of Bayesian estimation is in choosing appropriate prior distributions (McAllister and Kirkwood 1998, Punt and Hilborn 1997). If information is available regarding the potential value of a parameter, an informative prior can be developed. Expert knowledge and information from related species can be used, so long as the information used to develop the prior is completely independent from the data used in the analysis. For example, an assessment of a fish species could develop a prior for the intrinsic rate of increase r from the distribution of r values for all the known species in the same genus. If there is no available information about a parameter, an uninformative prior can be used. An uninformative prior conveys ignorance about the value of the parameter, so that only the likelihood function provides information about the parameter.
For the large coastal shark assessment, the demographic analyses were used to develop an informative prior for r in a Bayesian surplus production model (McAllister and Pikitch 1998a, 1998b; NMFS 1998). The demographic analysis was completely independent of the catch data, so it was a legitimate source of prior information. A˙Bayesian stock production model for large coastal sharks, using an informative prior on r generated the joint posterior on r and K shown in Figure 2, analogous to the joint likelihood surface in the classical assessment (Figure 1). Note that the range of most likely values of r and K is much more restricted.
Figure 2. Joint posterior of r and K, from McAllister and Pikitch 1998a.
|
The Bayesian method allowed the unrelated information from a demographic model and a stock production model to be combined, to increase the accuracy and precision of the estimates of r and K. The ability to incorporate prior information about demographics into the production model was probably the main reason why the Bayesian method was adopted for the most recent large coastal shark assessment (NMFS 1998).
Bayesian decision analysis
Estimating r and K is only part of the function of an assessment. Fisheries managers are required to decide on a management action, such as the total allowable catch (TAC) for large coastal sharks, when there is uncertainty about the state of nature (stock size relative to K and the value of r). An assessment must therefore calculate the probability of each competing hypothesis about the state of nature. Also, the consequences of each proposed management action must be calculated under each state of nature, and integrated across states of nature. The consequences of the management actions include such indicators of policy performance as the probability of stock recovery to K/2 within 10 years or the probability of the stock decreasing in the next 10 years (McAllister and Pikitch 1998b).
In Bayesian decision analysis, the posterior expected values and distributions of the indicators of policy performance are calculated through Monte Carlo simulation as follows (McAllister and Pikitch 1998b). A state of nature is drawn from the posterior pdf of the parameters from the assessment. These parameter values are used to calculate the stock size trajectory throughout the time series, with the catch in the future determined by the TAC being considered. This procedure is followed many times to determine the probability of the population increasing in the next ten years and other indicators. These results can be integrated across all the possible values of a parameter such as r, or calculated for several possible ranges of the parameter (McAllister and Pikitch 1998b). Because Bayesian decision analysis presents managers with decision tables and probability distributions instead of point estimates, management decisions can be made with an awareness of uncertainty.
Bayesian methods allow uncertainty to be formally incorporated into an assessment, and allow all available biological data to be included in the model. For many shark species there is little available fisheries data, so that including biological information in the form of priors will greatly improve the accuracy (and hence the usefulness) of the assessment models.
References
Dennis, B. 1996. Discussion: should ecologists become Bayesians? Ecological Applications 6(4): 1095-1103.
Hilborn, R., E. K. Pikitch and M. K. McAllister. 1994. A Bayesian estimation and decision analysis for an age-structured model using biomass survey data. Fisheries Research 19:17-30.
Hilborn R. and C. J. Walters. 1992. Quantitative fisheries stock assessment: choice, dynamics and uncertainty. Chapman and Hall. New York.
McAllister, M. K. and E.K. Pikitch. 1998a. A Bayesian approach to assessment of sharks: fitting a production model to large coastal shark data. Shark Evaluation Workshop 1998. SB-IV-26.
McAllister, M. K. and E. K. Pikitch. 1998b. Evaluating the potential for recovery of large coastal sharks: a Bayesian decision analysis. Shark Evaluation Workshop 1998. SB-IV-27.
McAllister, M. K., E. K. Pikitch, A. E. Punt and R. Hilborn. 1994. A Bayesian approach to stock assessment and harvest decisions using the sampling/importance resampling algorithm. Canadian Journal of Fisheries and Aquatic Sciences 51:2673-2687.
McAllister, M.K., and Kirkwood, G. P. 1998. Bayesian stock assessment: a review and example application using the logistic model. ICES Journal of Marine Science 55:1031-1060.
National Marine Fisheries Service. 1998. Report of the Shark Evaluation Workshop. National Oceanic and Atmospheric Association. National Marine Fisheries Service. Southeast Fisheries Science Center. 3500 Delwood Beach Road. Panama City, FL 32408.
National Marine Fisheries Service. 1996. Report of the Shark Evaluation Workshop. National Oceanic and Atmospheric Association. National Marine Fisheries Service. Southeast Fisheries Science Center. 75 Virginia Beach Dr. Miami, FL 33149.
Punt, A. E. and R. Hilborn. 1997. Fisheries stock assessment and decision analysis: the Bayesian approach. Reviews in Fish Biology and Fisheries 7:35-63.
Elizabeth A. Babcock and Ellen K. Pikitch,
Marine Conservation Programs, Wildlife Conservation Society, Bronx Zoo, Bronx, NY 10460, USA.
Email: bbabcock@wcs.org and epikitch@wcs.org
|
|
|
|
|
