Bayesian Mode Regression

Abstract

Like mean, quantile and variance, mode is also an important measure of central tendency and data summary. Many practical questions often focus on "Which element (gene or file or signal) occurs most often or is the most typical among all elements in a network?". In such cases mode regression provides a convenient summary of how the regressors affect the conditional mode and is totally different from other regression models based on conditional mean or conditional quantile or conditional variance. Some inference methods have been used for mode regression but none of them from the Bayesian perspective. This paper introduces Bayesian mode regression by exploring three different approaches. We start from a parametric Bayesian model by employing a likelihood function that is based on a mode uniform distribution. It is shown that irrespective of the original distribution of the data, the use of this special uniform distribution is a very natural and effective way for Bayesian mode regression. Posterior estimates based on this parametric likelihood, even under misspecification, are consistent and asymptotically normal. We then develop a nonparametric Bayesian model by using Dirichlet process (DP) mixtures of mode uniform distributions and finally we explore Bayesian empirical likelihood mode regression by taking empirical likelihood into a Bayesian framework. The paper also demonstrates that a variety of improper priors for the unknown model parameters yield a proper joint posterior. The proposed approach is illustrated using simulated datasets and a real data set.