Nesting success:

Younger male sparrows may or may not nest during a mating season, perhaps depending on their physical characteristics. Re-searchers have recorded the nesting success of 43 young male sparrows of the same age, as well as their wingspan, and the data appear in the file msparrownest.dat. Let Yi be the binary indicator that sparrow i successfully nests, and let xi denote their wingspan. Our model for Yi is logit Pr(Yi = 1|α, β, xi) = α +βxi, where the logit function is given by logit θ = log[θ/(1 — θ)].

a) Write out the joint sampling distribution  and simplify as much as possible.

b) Formulate a prior probability distribution over α and β by considering the range of Pr(Y = 1|α, β, x) as x ranges over 10 to 15, the approximate range of the observed wingspans.

c) Implement a Metropolis algorithm that approximates p(α, β|y, x). Adjust the proposal distribution to achieve a reasonable acceptance rate, and run the algorithm long enough so that the effective sample size is at least 1,000 for each parameter.

d) Compare the posterior densities of α and β to their prior densities.

e) Using output from the Metropolis algorithm, come up with a way to make a confidence band for the following function fαβ(x) of wingspan:

where α and β are the parameters in your sampling model. Make a plot of such a band.