Solve the following problem:
(Gelfand and Dey, 1994) Consider a density function f(x|θ) and a prior distribution π(θ) such that the marginal m(x) = ?Θ f(x|θ)π(θ)dθ is finite a.e.
The marginal density is of use in the comparison of models since it appears in the Bayes factor .
a. Give the general shape of an importance sampling approximation of m.
b. Detail this approximation when the importance function is the posterior distribution and when the normalizing constant is unknown.
c. Show that, for a proper density τ ,
