Are Bayesian Inferences Weak for Wasserman's Example?

Abstract

An example was given in the textbook All of Statistics (Wasserman, 2004, pp. 186–188) for arguing that, in the problems with a great many parameters Bayesian inferences are weak, because they rely heavily on the likelihood function that captures information of only a tiny fraction of the total parameters. Alternatively, he suggested non Bayesian Horwitz–Thompson estimator, which cannot be obtained from a likelihood-based approaches, including Bayesian approaches. He argued that Horwitz–Thompson estimator is good since it is unbiased and consistent. In this article, the mean square errors of Horwitz–Thompson estimator is compared with a Bayes estimator at a wide range of parameter configurations. These two estimators are also simulated to visualize them directly. From these comparisons, the conclusion is that the simple Bayes estimator works better than Horwitz–Thompson estimator for most parameter configurations. Hence, Bayesian inferences are not weak for this example.

Keywords:

• Bayes estimator
• Critique of Bayesian inference
• Decision theory
• Horwitz–Thompson estimator
• Wasserman's example

Mathematics Subject Classification:

• 62Axx
• 94A20
• 62D05

Acknowledgments

This work was supported by Natural Sciences and Engineering Research Council of Canada. The author thanks Weixin Yao for providing helpful comments on an earlier draft of this article.