A Bayesian multi-category kernel classication method is proposed. The hierarchical model is treated with a Bayesian inference procedure and the Gibbs sampler is implemented to find the posterior distributions of the parameters. The practical advantage of the full probabilistic model-based approach is that probability distributions of prediction can be obtained for new data points, which gives a more complete picture of classication. Large computational savings and improved classication performance can be achieved by a projection of the data to a subset of the principal axes of the feature space. The algorithm is aimed at high dimensional data sets where the dimension of measurements exceeds the number of observations. The applications considered in this paper are microarray, image processing and near-infrared spectroscopy data.