Exercise 318 EM

Show that iffo(p) is a beta probability density with parameters a and P and that the data are s successful searches in a attempts, then the posterior probability density for P obtained from Eq. (3.96) is also a beta density, with updated parameters a + s and P + a — s. Clark and Mangel (2000) provide further elaborations of how this updating result can be used in the prediction of patch leaving rules.

We close with one more consideration of that pesky problem of estimation of N and p for the binomial distribution, given k successes. If we assume that N is uniformly distributed (see Raftery (1988) for the case where N has a Poisson prior), that p has a beta density with parameters a and P and that the probability of k events is binomially distributed given p and N, then the posterior distribution for N and p, found in a manner analogous to Eq. (3.97) is proportional to (n)-Pk+a—1(1 — _p)N—k+P—1, which is nice and well behaved (e.g. we can find sensible maximum posterior estimates for both N and p, but we won't do that here - on to parasitoids!).

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