Bayesian Friends with Benefits: Partial Shrinkage

• Activity Number: 540
• Type: Invited
• Date/Time: Wednesday, August 1, 2012 : 2:00 PM to 3:50 PM
• Sponsor: Biometrics Section
• Author(s): Xiao-Li Meng*+
• Companies: Harvard University
• Keywords: Bayesian inferences ; Nuisance parameter ; Partially Bayes ; Sequentially partial shrinkage ; Simultaneously partial shrinkage

Abstract

Partially Bayes inference refers to a class of methods where a prior is used but only for the nuisance parameter. Such methods are appealing to those who are unwilling to seriously commit themselves to the whole Bayesian philosophy, yet they see some of its potential benefits and hence are willing to experiment, especially for the part of the inference that they want to get rid of. Using a simple normal example, this talk demonstrates that being partially Bayesian does come with its hoped for benefits in the form of partial shrinkage, namely, there is a "borrowing information" as with full Bayes methods except the amount of shrinkage is less. The loss of the full benefit persists even if one carries out partial shrinkage sequentially for all parameters including primary parameters, namely, by using partial prior for each parameter in any particular order. However, when partial shrinkage is carried out simultaneously for all parameters involved, the full shrinkage benefit is restored. These findings not only reveal the evolutionary link between Bayes and Partially Bayes, but also remind us that like most relationships in life, benefits come with attachments, obvious or hidden.