In Sec. 7.4, we introduced Bayes estimators. For simple loss functions, such as squared error and absolute error, we were able to derive general forms for Bayes estimators. In many real problems, loss functions are not so simple. Simulation can often be used to approXimate Bayes estimators. Suppose that we are able to simulate a sample θ(1), . . . , θ(v) (either directly or by Gibbs sampling) from the posterior distribution of some parameter θ given some observed data X = x. Here, θ can be real valued or multidimensional. Suppose that we have a loss function L(θ, a), and we want to choose a so as to minimize the posterior mean E[L(θ, a)|x].
a. Describe a general method for approximating the Bayes estimate in the situation described above.
b. Suppose that the simulation variance of the approximation to the Bayes estimate is proportional to 1 over the size of the simulation. How could one compute a simulation standard error for the approximation to the Bayes estimate?