@conference {412,
	title = {Estimates of constrained multi-class a posteriori probabilities in time series problems with neural networks},
	booktitle = {Proceedings of the International Joint Conference on Neural Networks},
	year = {1999},
	publisher = {IEEE, United States},
	organization = {IEEE, United States},
	address = {Washington, DC, USA},
	abstract = {<p>In time series problems, where time ordering is a crucial issue, the use of Partial Likelihood Estimation (PLE) represents a specially suitable method for the estimation of parameters in the model. We propose a new general supervised neural network algorithm, Joint Network and Data Density Estimation (JNDDE), that employs PLE to approximate conditional probability density functions for multi-class classification problems. The logistic regression analysis is generalized to multiple class problems with softmax regression neural network used to model the a-posteriori probabilities such that they are approximated by the network outputs. Constraints to the network architecture, as well as to the model of data, are imposed, resulting in both a flexible network architecture and distribution modeling. We consider application of JNDDE to channel equalization and present simulation results.</p>
},
	keywords = {Approximation theory, Computer simulation, Constraint theory, Data structures, Joint network-data density estimation (JNDDE), Mathematical models, Multi-class a posteriori probabilities, Neural networks, Partial likelihood estimation (PLE), Probability density function, Regression analysis},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0033325263\&partnerID=40\&md5=8c6134020b0b2a9c5ab05b131c070b88},
	author = {J I Arribas and Jes{\'u}s Cid-Sueiro and T Adali and H Ni and B Wang and A R Figueiras-Vidal}
}
@conference {411,
	title = {Neural architectures for parametric estimation of a posteriori probabilities by constrained conditional density functions},
	booktitle = {Neural Networks for Signal Processing - Proceedings of the IEEE Workshop},
	year = {1999},
	publisher = {IEEE, Piscataway, NJ, United States},
	organization = {IEEE, Piscataway, NJ, United States},
	address = {Madison, WI, USA},
	abstract = {<p>A new approach to the estimation of {\textquoteright}a posteriori{\textquoteright} class probabilities using neural networks, the Joint Network and Data Density Estimation (JNDDE), is presented in this paper. It is based on the estimation of the conditional data density functions, with some restrictions imposed by the classifier structure; the Bayes{\textquoteright} rule is used to obtain the {\textquoteright}a posteriori{\textquoteright} probabilities from these densities. The proposed method is applied to three different network structures: the logistic perceptron (for the binary case), the softmax perceptron (for multi-class problems) and a generalized softmax perceptron (that can be used to map arbitrarily complex probability functions). Gaussian mixture models are used for the conditional densities. The method has the advantage of establishing a distinction between the network parameters and the model parameters. Complexity on any of them can be fixed as desired. Maximum Likelihood gradient-based rules for the estimation of the parameters can be obtained. It is shown that JNDDE exhibits a more robust convergence characteristics than other methods of a posteriori probability estimation, such as those based on the minimization of a Strict Sense Bayesian (SSB) cost function.</p>
},
	keywords = {Asymptotic stability, Constraint theory, Data structures, Gaussian mixture models, Joint network and data density estimation, Mathematical models, Maximum likelihood estimation, Neural networks, Probability},
	doi = {https://doi.org/10.1109/NNSP.1999.788145},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0033321049\&partnerID=40\&md5=7967fa377810cc0c3e6a4d9020024b80},
	author = {J I Arribas and Jes{\'u}s Cid-Sueiro and T Adali and A R Figueiras-Vidal}
}
