(Smith and Gelfand, 1992) Show that when evaluating an integral based on a posterior distribution
where π is the prior distribution and the likelihood function, the prior distribution can always be used as an instrumental distribution.
a. Show that the variance of the weight is finite when the likelihood is bounded.
b. Compare the previous choice with choosing (θ|x) as the instrumental distribution when the likelihood is proportional to a density. (Hint: Consider the case of exponential families.)
c. Discuss the drawbacks of this (these) choice(s) in specific settings.
d. Show that a mixture between both instrumental distributions can ease some of the drawbacks.