In this paper, a new algorithm, the Joint Network and Data Density Estimation (JNDDE), is proposed to estimate the {\textquoteleft}a posteriori{\textquoteright} probabilities of the targets with neural networks in multiple classes problems. It is based on the estimation of conditional density functions for each class with some restrictions or constraints imposed by the classifier structure and the use Bayes rule to force the a posteriori probabilities at the output of the network, known here as a implicit set. The method is applied to train perceptrons by means of Gaussian mixture inputs, as a particular example for the Generalized Softmax Perceptron (GSP) network. The method has the advantage of providing a clear distinction between the network architecture and the model of the data constraints, giving network parameters or weights on one side and data over parameters on the other. MLE stochastic gradient based rules are obtained for JNDDE. This algorithm can be applied to hybrid labeled and unlabeled learning in a natural fashion.

}, keywords = {Generalized softmax perceptron (GSP) network, Joint network and data density estimation (JNDDE), Mathematical models, Maximum likelihood estimation, Neural networks, Probability density function, Random processes}, url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0033326060\&partnerID=40\&md5=bb38c144dac0872f3a467dc12170e6b6}, author = {Juan I. Arribas and Jes{\'u}s Cid-Sueiro and T Adali and A R Figueiras-Vidal} }