Bayesian nonparametric monotone regression

Abstract

In many applications there is interest in estimating the relation between a predictor and an outcome when the relation is known to be monotone or otherwise constrained due to the physical processes involved. We consider one such application-inferring time-resolved aerosol concentration from a low-cost differential pressure sensor. The objective is to estimate a monotone function and make inference on the scaled first derivative of the function. We proposed Bayesian nonparametric monotone regression, which uses a Bernstein polynomial basis to construct the regression function and puts a Dirichlet process prior on the regression coefficients. The base measure of the Dirichlet process is a finite mixture of a mass point at zero and a truncated normal. This construction imposes monotonicity while clustering the basis functions. Clustering the basis functions reduces the parameter space and allows the estimated regression function to be linear. With the proposed approach we can make closed-formed inference on the derivative of the estimated function including full quantification of uncertainty. In a simulation study the proposed method performs similar to other monotone regression approaches when the true function is wavy but performs better when the true function is linear. We apply the method to estimate time-resolved aerosol concentration with a newly developed portable aerosol monitor. The R package bnmr is made available to implement the method.

Dynamic Pricing with Volume Discounts in Online Settings

“…Second, the choice of Lognormal prior for the price basis ϕ u (•) forces such distributions to be non-negative, which in turn induces the monotonicity of the approximating function w.r.t. the price p (see Wilson et al (2020) for more details). As discussed before, this property is also commonly reflected by real demand functions and allows for fast learning of such curves.…”

Incorporating prior information into distributed lag nonlinear models with zero-inflated monotone regression trees

“…Some previous work has estimated monotone exposure-response functions for exposure to air pollution or weather and a health outcome assessed on the same day (Powell et al, 2012;Wilson et al, 2014), and there is a rich statistical literature on shape constrained regression for a general regression function. Approaches for shape constrained regression in a general regression setting include piecewise linear functions (Hildreth, 1954;Brunk, 1955), kernel smoothers Mammen (1991), a large number of spine based approaches (Ramsay, 1988;Neelon and Dunson, 2004;Meyer, 2008;Wang and Li, 2008;Meyer et al, 2011;Meyer, 2012;Powell et al, 2012) and Bernstein polynomial methods (Chang et al, 2005(Chang et al, , 2007Curtis and Ghosh, 2011;Wilson et al, 2014;Ding and Zhang, 2016;Wilson et al, 2020). Chipman et al (2021) proposed a monotone regression model based on the popular nonparametric Bayesian additive regression trees (BART) framework (Chipman et al, 2010) that constrains a general regression function to be monotone with respect to some or all predictors.…”