The summary statistics of the posterior distributions of r and A are shown in Tables 2-2 and
2-3. Further, the likelihood estimates using the BFGS algorithm were not consistent (i.e. they
failed to converge to the same estimates every time). The standard deviation of the estimates of r
and A increased with the increased variance in the prior gamma distribution set for A. Figure 2-1
shows the influence of the prior distributions on the posterior distributions with low sample
sizes.
Conclusions and Discussion
Small sample sizes (when R is less than 100 and T is less than 10) produce flat likelihood.
This makes likelihood-based estimation difficult. Computer algorithms like BFGS or Nelder-
Mead rely on smooth likelihood surfaces (Press et al., 1994) and also rely on computers capable
of high precision for parameter estimation with flat likelihood. The large standard errors
produced when using the Nelder-Mead algorithm is indicative of the flat likelihood surface. The
inconsistency in the results from the BFGS algorithm in parameter estimation is also indicative
of such a surface.
From the results in Tables 2-2 and 2-3, it may be inferred that Bayesian priors on A do play
an important role in the posterior distribution of the parameters when using the Gibbs sampler
algorithm. Hence from a biological standpoint, given low sample sizes, the choice of an
appropriate prior is critical to obtain meaningful estimates of animal abundance.
Considering that in this model A is the important parameter from a wildlife management
perspective and very difficult to estimate from field surveys, information obtained even from
small sample sizes would be helpful from a long term monitoring perspective. Bayesian
approaches do facilitate this process of updating parameter estimates on improved prior beliefs.