@article {409,
	title = {Cost functions to estimate a posteriori probabilities in multiclass problems},
	journal = {IEEE Transactions on Neural Networks},
	volume = {10},
	year = {1999},
	pages = {645-656},
	abstract = {<p>The problem of designing cost functions to estimate a posteriori probabilities in multiclass problems is addressed in this paper. We establish necessary and sufficient conditions that these costs must satisfy in one-class one-output networks whose outputs are consistent with probability laws. We focus our attention on a particular subset of the corresponding cost functions; those which verify two usually interesting properties: symmetry and separability (well-known cost functions, such as the quadratic cost or the cross entropy are particular cases in this subset). Finally, we present a universal stochastic gradient learning rule for single-layer networks, in the sense of minimizing a general version of these cost functions for a wide family of nonlinear activation functions.</p>
},
	keywords = {Cost functions, Estimation, Functions, Learning algorithms, Multiclass problems, Neural networks, Pattern recognition, Probability, Problem solving, Random processes, Stochastic gradient learning rule},
	issn = {10459227},
	doi = {10.1109/72.761724},
	url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0032643080\&partnerID=40\&md5=d528195bd6ec84531e59ddd2ececcd46},
	author = {Jes{\'u}s Cid-Sueiro and J I Arribas and S Urban-Munoz and A R Figueiras-Vidal}
}
