McLean (2003) classified koalas into one of nine possible tooth wear classes. A useful model for this kind of data is a multinomial distribution, where the chance of an individual being classified into a given class is a specified probability. The sum of these probabilities must be equal to one, because the classes define all possible classifications. The data are used to estimate these probabilities, which reflect the age structure of the population.

The Dirichlet distribution is a useful prior for the probabilities because it is a multivariate distribution (with one value for each class) and the sum of the probabilities is equal to one, as required for the multinomial analysis. It is the multivariate equivalent of the beta distribution. An uninformative Dirichlet distribution can be specified by setting all its parameters equal to one. See Appendix B for more information on the Dirichlet, multinomial, and other distributions.

In the analysis of the data on the koala age structure (Table 3.1), we will use an uninformative prior for the age structure. The WinBUGS code may be written as:

model

# number of koalas in each tooth wear class drawn from a multinomial distribution p[1:9] ~ ddirch(alpha[])

# uninformative prior for proportions (p[]) if all values of alpha are equal to one

The data may be entered as:

list(N = 3 97, Y = c(55, 132, 88, 48, 31, 26, 14, 3, 0), alpha = c(1, 1, 1, 1, 1, 1, 1, 1, 1))

The result of taking 100000 samples from the posterior distribution provides the predicted age structure of the koala population on Snake Island (Fig. 3.5).

different species is used to calculate diversity indices. For example, Shannon's diversity index is (Begon et al., 2005):

H Pi ln(P), i=i where S is the number of species in the community and Pi is the proportion of individuals in the community that belong to species i. We will never know the proportions precisely, so our estimate of Shannon's diversity index will be imprecise. Although it might be possible to propagate the uncertainty in some simple data transformations or by using re-sampling methods in a frequentist framework, this example is easy to analyse with Bayesian methods (Box 3.15).

Tooth wear class

Fig. 3.5 The predicted age structure of koalas on Snake Island in 1997 (McLean, 2003). The columns represent the means of the posterior distribution and the error bars represent 95% credible intervals.

IVB IVC

Fig. 3.5 The predicted age structure of koalas on Snake Island in 1997 (McLean, 2003). The columns represent the means of the posterior distribution and the error bars represent 95% credible intervals.

Box 3.15

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