Quantile Regression Neural Networks: A Bayesian Approach

TL;DR

This paper presents a Bayesian neural network approach for quantile regression using an asymmetric Laplace distribution, providing theoretical consistency and practical MCMC-based implementation.

Contribution

It introduces a novel Bayesian neural network method for quantile regression with proven posterior consistency under model misspecification.

Findings

Posterior distribution is asymptotically consistent.

The method performs well in simulation studies.

Real data examples demonstrate practical applicability.

Abstract

This article introduces a Bayesian neural network estimation method for quantile regression assuming an asymmetric Laplace distribution (ALD) for the response variable. It is shown that the posterior distribution for feedforward neural network quantile regression is asymptotically consistent under a misspecified ALD model. This consistency proof embeds the problem from density estimation domain and uses bounds on the bracketing entropy to derive the posterior consistency over Hellinger neighborhoods. This consistency result is shown in the setting where the number of hidden nodes grow with the sample size. The Bayesian implementation utilizes the normal-exponential mixture representation of the ALD density. The algorithm uses Markov chain Monte Carlo (MCMC) simulation technique - Gibbs sampling coupled with Metropolis-Hastings algorithm. We have addressed the issue of complexity associated with the afore-mentioned MCMC implementation in the context of chain convergence, choice of starting values, and step sizes. We have illustrated the proposed method with simulation studies and real data examples.