Shape restricted regression with random Bernstein polynomials

TL;DR

This paper introduces a Bayesian approach to shape restricted regression using priors on Bernstein polynomials, enabling flexible, smooth function estimation with efficient algorithms and demonstrated through simulation studies.

Contribution

It develops novel priors on Bernstein polynomials for shape restricted regression, allowing incorporation of geometric info and efficient MCMC algorithms.

Findings

Effective in estimating smooth shape-restricted functions

Comparable or superior to existing density-regression methods

Provides practical algorithms for prior and posterior generation

Abstract

Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only smooth functions, can easily incorporate geometric information into the prior, and can be generated without computational difficulty. Algorithms generating priors and posteriors are proposed, and simulation studies are conducted to illustrate the performance of this approach. Comparisons with the density-regression method of Dette et al. (2006) are included.