Consider the Bayes estimator of a binomial parameter π using a beta prior distribution.
a. Does any beta prior produce a Bayes estimator that coincides with the ML estimator?
b. Show that the ML estimator is a limit of Bayes estimators, for a certain sequence of beta prior parameter values.
c. Find an improper prior density (one for which its integral is not finite) such that the Bayes estimator coincides with the ML estimator. (In this sense, the ML estimator is a generalized Bayes estimator.)
d. For Bayesian inference using loss function w(θ)(T – θ)2, the Bayes estimator of θ is the posterior expected value of θw(θ) divided by the posterior expected value of w(θ). With loss function (T – π)2/[π(1 – π)], show the ML estimator of π is a Bayes estimator for the uniform prior distribution.
e. The risk function is the expected loss, treated as a function of π. For the loss function in part (d), show the risk function is constant. (Bayes’ estimators with constant risk are minimax; their maximum risk is no greater than the maximum risk for any other estimator.)
f. Show that the Jeffreys prior for π equals the beta density with α = β =.5.
