2. For this question, assume the likelihood function is as described in Section 2.2 with known error precision, h = 1, and x¡ = 1 for i = 1 ,..., N.

(a) Assume a Uniform prior for B such that B ~ U (@, y). Derive the posterior p(fly).

(b) What happens to p(8|y) as a -> -co and y -> 00?

(c) Use the change-of-variable theorem (Appendix B, Theorem B.21) to derive the prior for a one-to-one function of the regression coefficient, g(f), assuming that B has the Uniform prior given in (a). Sketch the implied prior for several choices of g() (e.g. g(B) = log(f), g(B) =

exp(B)

!+exp(B) . g

(f) = exp(f), etc.).

(d)

Consider what happens to the priors in part

(c) as of -> -00 and y -> 00.

(e)

Given your answers to part (d), discuss whether a prior which is 'nonin-

formative' when the model is parameterized in one way is also 'noninfor-

mative' when the model is parameterized in a different way.