Researchers in the fields of neural networks, statistics, machine learning, and artificial intelligence have followed three basic approaches to developing new pattern classifiers. Probability Density Function (PDF) classifiers include Gaussian and Gaussian Mixture classifiers which estimate distributions or densities of input features separately for each class. Posterior probability classifiers include multilayer perceptron neural networks with sigmoid nonlinearities and radial basis function networks. These classifiers estimate minimum-error Bayesian a posteriori probabilities (hereafter referred to as posterior probabilities) simultaneously for all classes. Boundary forming classifiers include hard-limiting single-layer perceptrons, hypersphere classifiers, and nearest neighbor classifiers. These classifiers have binary indicator outputs which form decision regions that specify the class of any input pattern. Posterior probability and boundary-forming classifiers are trained using discriminant training. All training data is used simultaneously to estimate Bayesian posterior probabilities or minimize overall classification error rates. PDF classifiers are trained using maximum likelihood approaches which individually model class distributions without regard to overall classification performance. Analytic results are presented which demonstrate that many neural network classifiers can accurately estimate posterior probabilities and that these neural network classifiers can sometimes provide lower error rates than PDF classifiers using the same number of trainable parameters. Experiments also demonstrate how interpretation of network outputs as posterior probabilities makes it possible to estimate the confidence of a classification decision, compensate for differences in class prior probabilities between test and training data, and combine outputs of multiple classifiers over time for speech recognition.