## Statistical Criterion and Parameter Estimation

The most widely used statistical approach for parameter estimation in fisheries stock assessment is the maximum-likelihood approach and the second most common and becoming more popular is Bayesian analysis. We will not discuss the Bayesian analysis here, other than to say that likelihood forms the foundation for Bayesian analysis. The concept of using likelihood and maximum-likelihood methods for estimating model parameters is to recognize that the data are known (we observed them) and the hypothesis is unknown (we do not know the model parameters). The notation or jargon that is used in likelihood methods is a simple statement such as:

L(data|hypothesis)

which reads as ''the likelihood of the data given the hypothesis.'' The data can be relative abundance indices, compositional information, tagging data, etc., and the hypotheses (plural) are the unknown model parameters which we are interested in estimating. The likelihood itself is simply a probability distribution (e.g., normal distribution, multinomial distribution, Poisson distribution, etc.).

To illustrate the concept of maximum-likelihood parameter estimation, consider the following 10 observations (the data denoted by Y)

Y, = (4.76 5.08 4.93 4.89 5.20 4.95 5.16 4.85 4.95 4.96)

Here we are interested in determining the mean and standard deviation of the data (i.e., the unknown parameters), and it is a fairly straightforward exercise to simply compute the mean and standard deviation (p = 4.973, u = 0.136). In a maximum-likelihood sense, we are interested in computing the likelihood of the data given hypotheses about various combinations of pm and um (where m is an index for alternative hypotheses). To do this, we must first specify the probability distribution from which these data were derived - for now let us assume they arose from a normal distribution. Given that the data were assumed to come from a normal distribution, the likelihood of the data given a hypothesis is computed using

The total likelihood of all of the data is then given by the product of eqn [13] applied to each of the 10 observations. This product of individual likelihoods raises a potential concern; if there are a large number of observations, then the product of several tiny numbers results in numbers that become so small that modern-day computers cannot distinguish this number from 0. Therefore, it is common practice to estimate parameters using log-likelihoods and simply sum the log-likelihoods for all observations. To estimate the most likely parameters that describes the data set, simply calculate the likelihood of the data over a systematic grid of values for pm and um and find the combination of pm and um values that maximizes the sum of log-likelihoods. Such a grid search approaches work

4.85

4.90

4.95 5.00 Mean

5.05

5.10

Figure 6 An example of a likelihood profile for assessing the odds of alternative hypotheses. The most likely value is indicated by the vertical dotted line, and the solid line represents the likelihood of alternative hypotheses about the population mean.

fine for small-dimensional problems (e.g., three unknown parameters). In higher-dimensional problems (e.g., >3 unknown parameters) it is usually much more efficient to use a quasi-Newton nonlinear search routine (e.g., Solver in the Microsoft Excel software).

The basic concept of computing the likelihood of the data given the parameter values and searching for parameter values that maximize the log-likelihood (either using a simple grid search or a nonlinear search algorithm) is the basic engine that is used in estimating model parameters for fisheries stock assessment. Assessing how well the model fits the data (or how well the parameters describe the data) is determined by the ratio of log-likelihoods. Moreover, uncertainty in the parameter estimates can be readily obtained by examining the likelihood profile (see example in Figure 6).

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