  Distance weighted discrimination (DWD) is a linear discrimination method that
is particularly well-suited for classification tasks with high-dimensional
data. The DWD coefficients minimize an intuitive objective function, which can
solved very efficiently using state-of-the-art optimization techniques.
However, DWD has not yet been cast into a model-based framework for statistical
inference. In this article we show that DWD identifies the mode of a proper
Bayesian posterior distribution, that results from a particular link function
for the class probabilities and a shrinkage-inducing proper prior distribution
on the coefficients. We describe a relatively efficient Markov chain Monte
Carlo (MCMC) algorithm to simulate from the true posterior under this Bayesian
framework. We show that the posterior is asymptotically normal and derive the
mean and covariance matrix of its limiting distribution. Through several
simulation studies and an application to breast cancer genomics we demonstrate
how the Bayesian approach to DWD can be used to (1) compute well-calibrated
posterior class probabilities, (2) assess uncertainty in the DWD coefficients
and resulting sample scores, (3) improve power via semi-supervised analysis
when not all class labels are available, and (4) automatically determine a
penalty tuning parameter within the model-based framework. R code to perform
Bayesian DWD is available at https://github.com/lockEF/BayesianDWD .
