Bayesian Methods

In the Bayesian approach, the experimenter believes from the very beginning that the unknown parameter ϑ is a random variable having its own probability distribution on the space Θ. A branch of statistics referred to as “decision theory” provides a formal structure to find optimal decision rules whenever possible. Berger treated modern decision theory with much emphasis on Bayesian arguments. In this chapter, the authors give a formal discussion of the prior and posterior distributions. They introduces the concept of conjugate priors and then posterior distributions are derived in some standard cases when ϑ has a conjugate prior, and also includes an example where the assumed prior for ϑ is not chosen from a conjugate family. The authors develop the point estimation problems under the squared error loss function and introduce the Bayes estimator for ϑ.