Solve the following problem:
Given a real importance sample X1,...,Xn with importance function g and target density f:
a. Show that the sum of the weights ωi = f(Xi)/g(Xi) is only equal to n in expectation and deduce that the weights need to be renormalized even when both densities have known normalizing constants.
b. Assuming that the weights ωi have been renormalized to sum to one, we sample, with replacement, n points X˜ j from the Xi's using those weights. Show that the X˜ j 's satisfy
n n
E [1/n ∑ h(X˜j)] = E [ ∑ ωih(Xi)]
j=i i=1
c. Deduce that if the formula above is satisfied for ωi = f(Xi)/g(Xi) instead, the empirical distribution associated with the X˜j 's is unbiased.