General Inequalities for Gibbs Posterior with Nonadditive Empirical Risk

• Activity Number: 364
• Type: Contributed
• Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Cheng Li*+ and Wenxin Jiang and Martin A. Tanner
• Companies: Northwestern University and Northwestern University and Northwestern University
• Keywords: Gibbs posterior ; GMM ; model selection ; oracle inequalities ; ranking ; risk minimization

Abstract

The Gibbs posterior is a useful tool for risk minimization, which adopts a Bayesian framework and can incorporate convenient computational algorithms such as Markov chain Monte Carlo. We derive risk bounds for the Gibbs posterior using some general nonasymptotic inequalities, which leads to nearly optimal convergence rates and optimal model selection to balance the approximation errors and the stochastic errors. These inequalities are formulated in a very general way that does not require the empirical risk to be a usual sample average over independent observations. We apply this framework to studying the convergence rate of the GMM (generalized method of moments) and deriving an oracle inequality for the ranking problem, where models are selected based on the Gibbs posterior with a nonadditive empirical risk.