Assign prior probabilities

The frog surveying problem (Chapter 1) provides an example of assigning priors to the different hypotheses. The two hypotheses are that the southern brown tree frog is present or absent from a surveyed site. If we had no previous information, then we might conclude there is nothing to choose between the two hypotheses before collecting data and assign equal prior probabilities to each. Such use of uniform priors is common in the face of ignorance.

However, if we know from previous surveys that the species is found in a particular fraction of ponds in the region, then we could use that fraction as the probability that the frog is present. The probability of the frog being absent is simply one minus the probability that it is present.

Finally, we might have a model for predicting the probability that the frog is present at ponds based on their characteristics, in which case this could be used as the prior. Each of these three cases reflects different levels of prior information. The first represents ignorance, the second the mean rate of occurrence of the frog within ponds, and the third how the rate of occurrence varies among ponds of different types. The addition of prior information in this way influences the results. If the pond has a high prior probability of the frog being present, then a single survey in which it is not seen would not be enough for us to be reasonably sure it is absent unless our ability to detect the frog was very good.

Using a uniform distribution to represent ignorance makes sense in some ways, but is problematic in others. Consider the case where we wish to determine the proportion of individuals that belong to each species in an African national park. Among the herbivores, we might be interested in the proportion of individuals that are zebras, wildebeest or some other species. By using the uniform distribution to represent ignorance, we would assign a one-third probability to each of these three classes (zebras, wildebeest, other). However, the probability of one-third is simply an artefact of our classification. The zebra would have had a probability of one-quarter if we had included gazelles as an additional class. Therefore, representing ignorance is not always straightforward.

Problems of representing ignorance also arise when specifying priors for continuous hypotheses (see also Box 3.12). For example, we may wish to estimate the density of territories of a species that are adjacent but non-overlapping. We could assign the prior distribution as uniform between 0.1 and 1.0 territories per ha if we were confident that the density was somewhere within that range but unsure of the actual value. This prior implies that the probability of the density being less than 0.2 territories per ha is 0.111 (0.1/0.9).

Alternatively, we could specify that the area of each territory is between 1 and 10 ha. This is equivalent to our limits for density (1.0 and 0.1 territories per ha, respectively). If we used a uniform distribution between 1 and 10 ha for territory size, the probability of the territory size being more than 5 ha is 0.555 (5/9), which is five times the probability calculated above for the equivalent density (0.2 territories per ha). Thus, we seem to have proved that 0.111 = 0.555.

The difference arises because the units of the two approaches are not linearly related, so the probabilities are not conserved when they are transformed (in this case by inversion). The prior distributions are very different (Fig. 2.2). This effect of the scale of measurement is not unique to Bayesian analyses. For example, a frequentist confidence interval based on territory size would not be equivalent to a confidence interval based on densities of territories per ha.

Density of territories per ha

Fig. 2.2 Prior distributions for the density of territories assuming that the density is uniformly distributed between 0.1 and 1.0 territories per ha (the uniform distribution) and assuming that the size of territories (the inverse of density) is uniformly distributed between 1 and 10 ha (the sharply peaked distribution). The probability of territory density being less than 0.2 per ha (the areas under the curves to the left of 0.2) is very different for the two priors.

Density of territories per ha

Fig. 2.2 Prior distributions for the density of territories assuming that the density is uniformly distributed between 0.1 and 1.0 territories per ha (the uniform distribution) and assuming that the size of territories (the inverse of density) is uniformly distributed between 1 and 10 ha (the sharply peaked distribution). The probability of territory density being less than 0.2 per ha (the areas under the curves to the left of 0.2) is very different for the two priors.

One of the difficulties in establishing prior probabilities is that humans tend to judge them poorly (Tversky and Kahneman, 1974; Kahneman et al., 1982; Ayton and Wright, 1994; Gigerenzer and Hoffrage, 1995; Anderson, 1998; Burgman, 2005). Construction of priors by using subjective judgement is likely to depend on a range of personal attributes, how the problem is presented, motivational biases and advocacy (Anderson, 1998; Burgman, 2005). Experts are not immune to these frailties of human nature (Burgman, 2005; see also Chapter 10).

Even when there are data for constructing priors, some subjective judgement is required to determine how the prior information is represented as a probability distribution. Frequentist methods are not free of subjective judgement because they also depend on judgements about the questions to be examined, how the data are collected, the variables to be analysed, and the statistical methods and models that are used (Howson and Urbach, 1991).

It could be argued that although science is not free of subjectivity (Burgman, 2005), it should seek to minimize it (Dennis, 1996). How can Bayesian methods be used reliably and convincingly in the face of subjectivity? One approach is to be as careful, rigorous and convincing in the choice of prior as in the collection of data. This book contains various examples of using previous results and data to construct priors. Furthermore, there will be many cases where the choice of prior has virtually no effect on the results.

However, there will still be cases where uncertainty in the choice of prior remains. This uncertainty can be regarded as an honest incorporation of subjectivity in science (Berger and Berry, 1988, Howson and Urbach, 1991). The role of science is to ensure that this opinion is updated logically as evidence accumulates. Bayes' rule ensures that beliefs are updated logically, with differing opinions converging as data are collected (Cox, 1946; Howson and Urbach, 1991; Crome et al., 1996; Jaynes, 2003).

There are extensions to Bayesian methods for dealing with uncertainty in the choice of priors. The methods, lumped under the title of 'robust Bayesian analysis', can also deal with uncertainty in the models used to represent the hypotheses. They involve, for example, placing bounds on the possible parameters or distributions for priors and likelihoods and, therefore, bounds on the possible posterior distributions (e.g. Berger, 1985; Walley, 1991; Ferson, 2005). Robust Bayesian methods are not without controversy, and they usually add to the computational burden. In providing an introduction to Bayesian methods for ecology, I will only touch on them briefly in Chapter 10. Interested readers are referred to Berger (1985) and Ferson (2005).

Although the prior can pose difficulties for Bayesian methods, it is in fact one of its strengths. Ecologists, in the discussion sections of journal articles, routinely consider their results in the light of previous studies. Bayesian methods provide a formal basis for these comparisons through the use of priors. Scientists may be forced to be more rigorous and less subjective when using priors to represent previous work than when simply using their judgement to make comparisons. Bayes' rule provides the means of incorporating previous findings into the formal interpretation of new data.

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