A. Status of the Spectacled Eider - U.S. Fish and Wildlife Service

S 1(t) = m1 CV(N(t)) N(t) (2.0)
where
= the CV multiplier for survey I,
CV(N1(t)) = the estimated survey precision (CV) ofabundance estimate 141(t).
Note that parameter values that maximize equation 4 are the maximum likelihood estimates, a
common point estimator in frequentist statistics. In Bayesian statistics, rather than maximizing
equation 4 we need to integrate the product ofit and the prior distribution ofthe parameters
(defined below).
Until fairly recently, most Bayesian analyses were restricted to cases where the prior distribution
could be chosen so that it was “conjugate” to the likelihood distribution, resulting in their product
being a distribution ofa known form, which was integratable by analytic methods. Advances in
computing power and numerical and Monte Carlo integration methods have removed this
restriction. We use the Sampling-Importance-Resampling routine ofRubin (1988), which Smith
and Gelfand (1992) advocate as a particularly useful and simple integration technique for
Bayesian statistics. In this method, values for the parameters are randomly selected from their
joint prior distribution to form a sample set 0~. The likelihood ofthe data given this particular O~
is calculated and stored. This is repeated, generating n1 O,’s with associated likelihoods. These n1
O,’s are then re-sampled n2 times with replacement, with probability equal to weight q~, where
L(011x) (12.)
j~1 L(831x)
Rubin (1988) showed that this generates a random sample from the joint posterior distribution of
size n2. The resampling with replacement from the n1 O,’s with weight CL makes this process
analogous to a weighted bootstrap procedure. We set values for n1 and n2 to yield smooth
posterior distributions, which depended on the number ofparameters estimated. For single survey
analyses, n1 = 300,000, n2 = 5,000. For analyses ofall three surveys, n1 = 8,000,000, n2 = 10,000.
For analyses ofall surveys with environmental stochasticity (equation 1), n1 = 25,000,000, n2 =
10,000. With the exception ofinitial population size, prior distributions were Uniform
distributions that were iteratively set to encompass the values of the posterior distribution for each
parameter.
For the full analysis using the deterministic population model (equation 1) and all three data sets,
values were randomly drawn from their prior distributions for the seven parameters (r, N0 ma, 1%,
a0~ and ar)) to form a 01. Those parameter values were then used to project a model
population trajectory using equation 1, forming a series ofmodel population sizes N1, N2, . .N~.
The likelihood ofthat population trajectory given the data was then calculated using equation 9,
and the 0, ‘s and associated likelihood value were stored. This was repeated n1 times to form the
Appendix II- Page 4