4.1 General issues
4.2 Expert opinion
4.3 Data summaries/meta-analysis
4.4 Default options for parameters and their priors
Developing prior distributions is undoubtedly the most controversial aspect of any Bayesian analysis (Lindley, 1983; Walters and Ludwig, 1994) although this needs to be balanced by the fact that not having to choose priors in other types of analyses is not necessarily uncontroversial either. Considerable care should be taken when selecting priors and the process by which priors are selected must be documented carefully. This is because inappropriate choices for priors can lead to incorrect inferences. We have noticed a tendency for analysts to underestimate uncertainty when specifying priors, and hence to specify unrealistically informative priors. For example, it is common to use priors that assign zero probability outside of some range (e.g. a uniform prior). However, this should be done with care because if some value is assigned zero probability a priori, Bayes theorem (Equations 1.5 and 1.6) ensures that the posterior distribution also assigns it zero probability. If the likelihood favours values close to where the prior assigns zero probability, the mode of the posterior may be at its extreme, which is not desirable. We suggest that the prior should assign a non-zero (but possibly very small) probability to all plausible values. Unfortunately, this can lead to extremely long run-times when using, for example, the grid search method (Section 2.1) for computing the posterior.
In many situations, it is not at all obvious which prior is the most appropriate and we suggest that the sensitivity of the results to the choice of the prior be examined and, if necessary, the implications for management reported to the decision makers.
There are two types of priors: informative and noninformative (or "reference"). Box and Tiao (1973) define a noninformative prior as one that provides little information relative to the experiment - in this case the stock assessment data. Informative prior distributions, on the other hand, summarise the evidence about the parameters concerned from many sources and often have a considerable impact on the results.
Using informative prior distributions allows the incorporation of information available to stock assessment scientists from the literature and in light of their experience with other stocks. However, using informative priors may lead to problems because of the subjective beliefs of stock assessment scientists. Unfortunately, even if we wanted to use noninformative priors, the best method for choosing such priors is still an issue of considerable debate. Uniform priors are generally chosen when noninformative priors are needed (Punt and Hilborn, 1997). Indeed, several of the examples in Section 3 assign uniform priors to model parameters. However, the use of noninformative priors is controversial because the results of the assessment may be sensitive to the choice of how the non-informative prior is specified (e.g. should q be assumed to be uniform or should perhaps the logarithm of q be assumed to be uniform).
In almost any stock assessment any prior will be informative with respect to some of quantities of interest in the model even if it is noninformative for others. When the results are sensitive to the choice of noninformative priors, attention needs to be given to the data included in the analysis because the likelihood is not very informative relative to the prior information (Kass and Wasserman, 1996). In such a situation, it is questionable whether there is much advice that can be provided to the decision makers. A key diagnostic when applying Bayesian methods is therefore to compare the posterior distributions for the primary model outputs with their prior distributions (e.g. spreadsheet EX3A.XLS). If the prior is very similar to the posterior, it often implies that the data provide very little information about the values of the model parameters.
We advocate a pragmatic approach to the choice between noninformative and informative priors and have, on occasion, incorporated both types into analyses (e.g. McAllister et al., 1994; Punt and Kennedy, 1997; Section 3.5). It is preferable to select a noninformative prior and test sensitivity to alternatives than to "dream up" an informative prior that perhaps markedly biases the results. On the other hand, well thought out informative priors can reduce uncertainty considerably. Indeed, using noninformative priors implies that no information from fisheries science is relevant to the parameter in question.
It is often easier to develop a prior for each parameter in turn (marginal priors) than to develop a joint prior for all of the parameters simultaneously. In fact, most of the Bayesian stock assessments to date have made the assumption that the priors for the various parameters are independent (e.g. McAllister et al., 1994; McAllister and Ianelli, 1997; Punt and Walker, 1998; Punt and Butterworth, 2000). However, care needs to be taken when specifying marginal priors to check whether, in fact, the (implicit) assumption that the parameters are independent is valid. For example, in its development of a prior distribution for the assessment of the Bering-Chukchi-Beaufort Seas stock of bowhead whales, Balaena mysticetus, the Scientific Committee of the IWC (IWC, 1995) considered each of the model parameters separately except for those which determine juvenile and adult natural mortality. It is clear that these two parameters cannot be independent because juvenile natural mortality must be higher than that for adults. Even though considerable care went into selecting the priors for the bowhead assessment, Butterworth (1995) has criticised them. This is because the priors for the age-at-maturity and the survival rate for adults were assumed to be independent when it would be anticipated from studies for other species (e.g. Gunderson and Dygert, 1988) that higher values of age-at-maturity are likely to be linked to higher values of adult survival.
We advocate parameterising the model with a single parameter to scale the population and selecting the remaining parameters so that they do not scale with population size. These remaining parameters are then comparable among stocks, making it considerably easier to construct priors. For example, it seems reasonable to assume that the prior distributions for B_{0} and r of a Schaefer model are independent of each other (the productivity of the population should not depend a priori on the size of the population prior to harvesting). In contrast, it is unreasonable to assume that the parameters conventionally used to parameterise a stock recruitment relationship, a and b are independent. Often, however, the choice of quantities on which to place priors is not simple. For example, should you place a prior on the initial or the final stock size? When the data are sufficiently informative, this choice is irrelevant. However, this will not always be the case. Our advice is that one should select parameters for which it is easy to specify priors (e.g. r and B_{0} rather than say MSY and current stock size) and which reduce computation time. For example, if a prior is placed on current population size, it is necessary to run the model "backwards" (Butterworth and Punt, 1995) while this is not the case if a prior is placed on the initial stock size. Running models "backwards" so that the current population size equals a pre-specified value can be very complicated as it usually involves solving a complicated non-linear equation.
A particular problem arises with the specification of priors for residual variances. It is often assumed that these variances are known without error (see, for example, EX4A.XLS). However, when allowance is made for uncertainty in these variances, this uncertainty can be shown to be substantial. For example, the spreadsheet EX4C.XLS includes a marginal posterior for the process error about the stock-recruitment relationship for Skeena River sockeye salmon. This posterior distribution is wide and assuming a single value for the process error variance would have lead to gross underestimation of uncertainty. Unfortunately, allowing the residual variances to be uncertain can increase the time required to compute Bayesian posteriors substantially.
The following two sections outline the two techniques used most frequently to develop informative prior distributions and the final section provides some advice on default choices for priors when applying typical methods of fisheries stock assessment.
In principle, one of the most powerful methods for developing informative priors is to synthesise the information from a group of experts. For example, International Whaling Commission (1995) developed priors for the assessment of the Bering-Chukchi-Beaufort Seas stock of bowhead whales by consensus.
Although the development of priors by consensus risks all the problems related to the impact of the subjective biases of the various parties in the assessment process (arguably priors developed using expert opinion are examples of "dreamt up" priors, to use an expression we used in the previous section), this approach can be successful. For example, as part of the assessment of the school shark, Galeorhinus galeus, resource off Australia, it was necessary for the assessment group to develop a prior distribution for the ratio of MSY to B_{MSY} (MSYR). This was achieved by tabulating estimates of MSYR for several other shark species (based on model estimates of increase rates at low population size) and asking each member of the assessment group to provide his/her prior distribution for MSYR for school shark. The members of the assessment group were provided with the values for other biological parameters (growth, natural mortality, etc.) for all of the species concerned. Although the priors suggested by the scientific members of the assessment group tended to be more pessimistic than those suggested by the industry members (Punt and Walker, 1998), this was nevertheless generally regarded as a successful attempt at specifying a prior. A similar exercise was conducted by the assessment group for Australia's eastern stock of gemfish (Smith and Punt, 1998).
A potentially major problem with the development of priors by consensus is that different "experts" will suggest different priors. It is far from a trivial exercise (theoretically) to pool such priors to form a "consensus prior" (and it is impossible to include more than one prior for each parameter in a Bayesian assessment). Unfortunately, relatively little work has been directed recently at this problem (Raftery et al., 1996). We recommend that the various priors be multiplied together and then normalised because at least this procedure has the desirable property that the assessment results are independent of whether the priors are pooled and then the assessment conducted or whether assessments are conducted using each alternative prior in turn and the results then pooled. One very undesirable feature of this approach to pooling, however, is that if one expert believes that some parameter value/model has zero probability, the posterior is forced to be consistent with this opinion. Therefore, if this approach is to be used, our earlier advice that no plausible value for a parameter should be assigned zero probability should be followed.
If the parameters of the stock assessment model are chosen to be independent of the parameter that scales the population, data for other species and stocks can be used to construct priors for the species for which an assessment is needed. This approach to conducting priors is known as meta-analysis. Methods for constructing priors using data for other stocks and species range from simply tabulating the estimates to hierarchical meta-analysis (Gelman et al., 1995; Hilborn and Liermann, 1998). Simple tabulation methods can be extended by fitting a smooth functional form to the data and by weighting each estimate by a measure of its uncertainty and comparability to the stock and species for which an assessment is required. For example, McAllister and Ianelli (1997) developed a prior for the slope at the origin of the stock-recruitment relationship, a, for the stock of yellowfin sole in the eastern Bering Sea by fitting a probability density function to estimates of a for several stocks of flatfish. Hierarchical meta-analysis (Gelman et al., 1995) is a more formal method for developing a prior for a parameter from values for that parameter for other stocks under the assumption that the stocks differ in that parameter.
Parameters such as steepness, h, can be computed for many stocks using, for example, the data compiled by Myers et al. (1995) and a prior constructed by selecting estimates for stocks that are "similar" to the one under consideration (e.g. Myers et al., 1999a). Care must, of course, be taken to specify how the stocks used were selected and to consider only those stocks for which the assessments are "reasonable" although what this is, is of course, subjective (Hilborn and Liermann, 1998).
"Selection bias" is a potential problem when developing a prior using data for similar stocks and species. Assessments in the literature tend to be for large productive populations (small, less productive populations in general receiving less research funding). If the stocks considered are not representative of all similar stocks, an inappropriate prior may be selected. For example, it has been argued that some of the priors used in the assessment of the Bering-Chukchi-Beaufort Seas stock of bowhead whales were inappropriate (Butterworth, 1995). This is because they were based on inferences for other species even though no other species of baleen whale are thought to be as long-lived as the bowhead.
We expect that as the use of Bayesian methods for stock assessment becomes more common, more effort will be directed towards collating and analysing data for a variety of stocks and species (e.g. natural mortality - Pauly, 1980; depensation - Liermann and Hilborn, 1997; whether catch rate is proportional to abundance - Dunn et al., 2000; steepness - Myers et al., 1999a). We hope that, in the future, informative priors will be available for most (if not all) of the parameters commonly used in fisheries stock assessments.
4.4.1 Biomass dynamics models
4.4.2 Age-structured/delay-difference models
4.4.3 Stock-recruitment models
The model parameters for stock-recruitment, biomass dynamics and age-structured assessments are usually the same for each application (even though the data available for assessment purposes may differ quite markedly among applications). In this section, we list the parameters for typical applications of these stock assessment methods along with default approaches to developing priors for the estimable parameters. The suggestions listed below are only defaults and when additional prior information is available for a given species, it should, of course, be used in preference to our suggestions.
Biomass dynamics models typically have three parameters (the pre-exploitation equilibrium biomass, B_{0}, the intrinsic rate of growth, r, and a catchability coefficient for each relative abundance index, q)[12]. The default prior for B_{0} is usually one that is uniform over some appropriate interval (say 0 to 10 times the cumulative catch). However, because the outcome of assessments and hence decision analyses can be very sensitive to the prior assumed for B_{0}, we recommend that analysts consider sensitivity to this prior; for example, by placing a uniform prior on the logarithm of B_{0}. Several studies have attempted to collate estimates of r (e.g. FishBase - Froese and Pauly, 1997). A default prior would be uniform over an interval that includes the estimate of r for all similar species. The default prior for q should be uniform on a log scale. This prior can be shown to be uninformative for B_{0} (Pikitch et al., 1993).
Priors can be placed on the variation of the observation errors (s in Equation 2.3) and it is possible, for some forms for this prior, to integrate across its prior analytically (e.g. Walters and Ludwig, 1994). However, most studies (e.g. McAllister et al., 1994; Section 2 of this manual) are based on pre-specifying the values for the s s.
Age-structured (e.g. Fournier and Archibald, 1982; Deriso et al., 1985; Hilborn, 1990b; McAllister and Ianelli, 1997) and delay-difference (Deriso, 1980; Schnute, 1985) models typically have far more parameters than do biomass dynamics models. Three of these are typically B_{0}, q and , and we recommend that the defaults listed in Section 4.4.1 be used. Apart from B_{0} and q, the most important parameter of an age-structured/delay-difference model is the parameter that determines how resilient recruitment is to changes in spawner stock size. This parameter is usually either the steepness of the stock-recruitment relationship, h (the fraction of virgin recruitment at 20% of B_{0}), or the slope of the stock-recruitment relationship at the origin, a. Meta-analyses have been conducted for both of these parameters (Myers et al., 1999a) and analysts should use the results of these analyses to select priors for their species.
Many age-structured stock assessments allow recruitment to differ from the value expected from the stock-recruitment relationship. The typical representation of this in a stock assessment model is:
(4.1)
where
R_{t} is the number of 0-year-olds during year t,The prior distribution most commonly assumed for d_{t} is log-normal with mean 1 and coefficient of variation s_{r}, where the value for s_{r} is pre-specified based on previous studies (for example, Beddington and Cooke, 1983; Myers et al., 1995; and Myers et al., 1999a).R() is the stock-recruitment relationship,
S_{t} is the spawner stock size at the start of year t, and
d_{t} is a multiplicative factor to adjust the output of the stock-recruitment relationship for year t.
The remaining parameters of most age-structured assessments are the probability of being mature as a function of age, weight-at-age, the vulnerability vector (often assumed to be a function of gear-type and sex), and the rate of natural mortality, M. For most species, there is usually sufficient information available on maturity to pre-specify the maturity schedule so no prior is needed. Similarly, there is usually sufficient data to simply pre-specify weight-at-age, although it should noted that it has been shown in several cases that the common assumption that weight- and maturity- at-age are independent of time is invalid (e.g. Clark et al., 1999). However, except in unusual circumstances, it is necessary to estimate M and vulnerability as a function of age/length using the assessment data. The prior assumed for M is usually uniform over a range selected based on the longevity of the species (e.g. U[0.15, 0.25yr^{-1}] for gadoids). Vulnerability is either modelled as the relative probability of capture by age (an individual parameter for each age) or using a smooth function (typically a logistic curve). For the first of these cases, the logarithm of selectivity is usually assumed to be uniformally distributed whereas in the latter cases, the ages-at-50%- and 95-selectivity can be assumed to be uniformally distributed (e.g. Smith and Punt (1998)).
The suggestions above are for typical age-structured stock assessments. Recently, however, assessments based on age-structured models have become increasingly tailored to the species in question. For example, Smith and Punt (1998) incorporate a prior on the extent of depensation in the stock-recruitment relationship (based on the meta-analysis by Liermann and Hilborn (1997)), allow for inter-annual correlation in d_{t} and place a (uniform) prior on the extent of density-dependence in selectivity. Several age-structured methods of stock assessment (e.g. ADAPT - Gavaris, 1988) do not directly include a stock-recruitment relationship but rather treat the strength of each year-class as a separate parameter. For such situations it is common to assume that the magnitude of a year-class or its logarithm is uniformly distributed (the latter being more common).
There are generally three parameters in an analysis of stock and recruitment data: (a) the slope of the curve at the origin, (b) the maximum number of recruits produced, and (c) the variability about the curve. These parameters apply both when stock-recruitment models are applied to species such as Pacific salmon (see Section 3.2), and also to the stock-recruitment relationships that form part of an age-structured model. The Ricker form of the stock-recruitment relationship is often written as:
(4.2)
The slope of the curve at the origin (exp(a)) is closely related to fecundity multiplied by survival to the age of recruitment, and the sustainable harvest rate will be proportional to this parameter. The parameter b determines the carrying capacity of the population.
Myers et al. (1999a) have analysed stock and recruitment data for hundreds of fish stocks[13] and conducted a meta-analysis for the slope of the stock-recruitment relationship at the origin (summarised in Table 4.1). They found that when the stock-recruitment curve is standardized so that spawners and recruits are in the same units, there is considerable consistency in this parameter across fish stocks.
Table 4.1: Mean and between-population standard deviation for four species groups of the logarithm of the slope of the stock-recruitment relationship at the origin, a (source: Table 1 of Myers et al. (1999a)).
Species-group |
Mean |
Standard deviation |
Clupeidae |
1.06 |
1.16 |
Gadidae |
1.01 |
0.51 |
Pleuronectidae |
0.79 |
0.34 |
Salmonidae |
1.43 |
0.18 |
Freshwater rearing may be limiting for salmon as well, and priors on carrying capacity based on summer or winter rearing area for coho salmon in streams could be constructed, as could the lake rearing capacity for sockeye salmon. For marine fishes, trophic or area based calculations could be used to formulate priors for carrying capacity. However, this approach is unlikely to be able to place tight bounds on carrying capacity. Unfortunately, the key when dealing with carrying capacity is to bound the high end of the range using basic biological knowledge because the data often do not provide this information.
Priors may prove very important in an analysis of stock and recruitment data. For example, for many stocks there is no information on the slope at the origin if there are few (or no) data at small spawner stock size. The fits to such data sets often result in biologically unrealistic estimates of the resilience of the population. In such cases it is important to use priors on the initial slope to make sure the conclusions are biologically reasonable. Other data sets may provide no information about the maximum recruitment level or the carrying capacity. The best fits to these data are often a straight line through the origin. In such cases the data provide no information on whether the stock could potentially be 10 or 10,000 times larger than the largest size ever observed. Here again, a prior becomes important to keep the analysis within biologically reasonable limits.
There is less consensus in the literature on how to specify the prior for the third parameter (the variability about the stock-recruitment relationship). Many studies have assumed that the value of this parameter is known (e.g. Smith and Punt, 1998) while others (e.g. McAllister et al., 1994) have used priors based on a meta-analysis. Other approaches include assuming an inverse-gamma prior or that the prior probability of a value of s is proportional 1/s^{n} (Walters and Ludwig, 1994; Punt and Butterworth, 1996).