  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.
