Multivariate Bayesian Convex Regression

• Activity Number: 61
• Type: Topic Contributed
• Date/Time: Sunday, August 4, 2013 : 4:00 PM to 5:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Lauren Hannah*+ and David B. Dunson
• Companies: Columbia University and Duke University
• Keywords: Convex Regression ; Bayesian Methods ; Nonparametric

Abstract

Convex optimization solution methods only work when objective functions and constraints are fully known. In many settings like reinforcement learning and approximate dynamic programming, these need to be estimated from data. The estimates need to maintain convexity so they can be used in conjunction with a solver. We propose a new Bayesian nonparametric multivariate method for regression subject to convexity constraints. It characterizes the unknown regression function as the max of a random collection of unknown hyperplanes. The Bayesian method offers multiple benefits over frequentist alternatives: model averaging produces estimates that are stable in an optimization setting, posterior sampling can be used to approximate distributional constraints for robust optimization, and the modularity of Bayesian methods easily extends this model to partially shape constrained settings.