## Annual mortality of powerful owls

Powerful owls are Australia's largest owl, with the mass of females averaging approximately 1.35 kg. However, as with many owls, they are difficult creatures to study. Powerful owls occur at low population densities, are mainly active at night, and are unreliably detected using typical owl survey methods (Wintle et al., 2005a). Because of this, few people have the necessary fortitude to study them. Approximately 30 powerful owl nestlings were banded in Victoria up to 1999, but none of these banded individuals were resighted as adults (McCarthy et al., 1999).

McCarthy et al. (1999) used data from three birds that could be identified without bands to estimate annual mortality of powerful owls. One of these birds disappeared after eight years, while the other two were observed to remain alive for ten and 17 years. A single death in one year and 35 years of survival suggests the annual mortality rate is 0.03 (1/36), although given the paucity of data it is not surprising that the 95% credible interval is wide (0.0066—0.14, using x = 1 and n = 36 in Box 3.11). This interval corresponds to the owls having an average lifetime of as little as approximately seven years (1/0.14) and approximately 150 years (1/0.0066) because average lifespan is approximately equal to the inverse of the annual mortality rate. The wide confidence interval is not surprising given the relatively small amount of data.

This upper limit of 150 years appears unreasonably large. The lifespan of a single owl, let alone the average of a species, is unlikely to be as long as 150 years. Although the credible interval for mortality of

0.0066—0.14 (using a uniform prior between zero and one) is consistent with the data, it is not consistent with common sense. It is tempting to limit the survival rate to sensible values. However, it would be good to do this in a repeatable, explicit and logically consistent way. This is where Bayesian statistics can lend a hand, by using additional information to set the prior.

One approach to establishing the prior would be to use intuition to limit the possible values. Given that this is a large bird, it is unlikely that the mortality rate is greater than about 0.2, which would correspond to an average lifespan of 5 years. Similarly, it is unlikely that the average lifetime is greater than 50 years, which corresponds to an annual mortality of 0.02. Using a prior for the annual mortality that is uniform between 0.02 and 0.2 leads to a posterior distribution with a mean of 0.06 and 95% credible interval of 0.02—0.14 (Box 3.12).

In some ways, this is a perfectly good estimate. The assumptions have been clearly stated and the data have been used in a logical way to update the prior belief by using Bayes' rule. Therefore, the estimate is internally consistent with the stated logic and data. However, if the same ecologist does this calculation on a different day, he or she may arrive at a different result by deciding that 40 years would be the maximum possible average age, or 4 or 10 years as the minimum. A different ecologist is likely to use a different prior again.

The exact same ecologist using the same intuition could arrive at a different answer by using a different formulation for the prior. Instead of assuming that the mortality rate had a prior that was uniform between 0.02 and 0.2, the ecologist might assume that the average age of death was uniform between 5 and 50 years. In this case, the prior for the mortality rate would not be uninformative, but peaked near 0.02 (Box 3.12).

The posterior distribution for the mortality rate is influenced by this prior, giving a mean of 0.037 and a 95% credible interval of 0.02 to 0.09. The difference occurs because the inverse of a uniformly distributed variable will be markedly skewed (e.g. Fig. 2.2).

The potential arbitrary choice of priors is one of the limitations of Bayesian statistics. However, the potentially arbitrary influence of subjective judgement on the interpretation of results is not limited to Bayesian statistics. Faced with a conclusion that the mean lifetime of powerful owls is likely to be between 7 and 150 years, most ecolo-gists would believe that the upper limit is unrealistically high, and would conclude that the mean lifetime is certainly less than 150 years.

Box 3.12