7.28 A simple probability model for the data of Table 7.1 has two probabilities as parameters, p, the probability of capture in any year and φ, the annual dipper survival probability. For the first five rows of the table the data are multinomial, with first two multinomial probabilities, respectively,

φp and φ2p(1 − p).

Describe fully the probability distribution for the 80 dippers released in 1984, noting that the study stopped in 1987. Figure 7.12 contains several panels. In one is shown the likelihood for this application. In two are shown the marginal posterior distributions for each of the parameters, corresponding to independent U(0, 1) prior distributions for the two parameters.

Explain in detail how the marginal distributions are obtained. In the last two figure panels are shown histograms resulting from simulations from the two marginal posterior distributions. The simulations have been obtained by means of an MCMC procedure, the Matlab code for which is shown in the program below.