Bayesian nonparametric multivariate convex regression

Lauren A. Hannah; David B. Dunson

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arXiv · 1109.0322

Bayesian nonparametric multivariate convex regression

Lauren A. Hannah·David B. Dunson

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Abstract

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 R^p, 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.

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BibTeX RIS

Lauren A. Hannah, David B. Dunson. 2011-09-01. Bayesian nonparametric multivariate convex regression. https://arxiv.org/abs/1109.0322

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