Bayesian Discriminant Analysis Using a High Dimensional Predictor

Abstract

We consider the problem of Bayesian discriminant analysis using a high dimensional predictor. In this setting, the underlying precision matrices can be estimated with reasonable accuracy only if some appropriate additional structure like sparsity is assumed. We induce a prior on the precision matrix through a sparse prior on its Cholesky decomposition. For computational ease, we use shrinkage priors to induce sparsity on the off-diagonal entries of the Cholesky decomposition matrix and exploit certain conditional conjugacy structure. We obtain the contraction rate of the posterior distribution for the mean and the precision matrix respectively using the Euclidean and the Frobenius distance, and show that under some milder restriction on the growth of the dimension, the misclassification probability of the Bayesian classification procedure converges to that of the oracle classifier for both linear and quadratic discriminant analysis. Extensive simulations show that the proposed Bayesian methods perform very well. An application to identify cancerous breast tumorbased on image data obtained using find needle aspirate is considered.