2011/09/01 by Lauren A. Hannah, David B. Dunson, Hannah, Lauren A. +1 · 1 citation
Computer Science · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Methodology (stat.ME) #Optimization and Control (math.OC) #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1109.0322
openalex publication_date 2011/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f subject to a convexity constraint. For example, in sequential decision processes the value of a state under optimal subsequent decisions may be known to be convex or concave. We propose a new Bayesian nonparametric multivariate approach based on characterizing the unknown regression function as the max of a random collection of unknown hyperplanes. This specification induces a prior with large support in a Kullback-Leibler sense on the space of convex functions, while also leading to strong posterior consistency. Although we assume that f is defined over Rp, we show that this model has a convergence rate of log(n)-1 n-1/(d+2) under the empirical L2 norm when f actually maps a d dimensional linear subspace to R. We design an efficient reversible jump MCMC algorithm for posterior computation and demonstrate the methods through application to value function approximation.