utes. Nonlinear regression curves can be also derived from nonparametric approximation techniques. Nonparametric methods present slower convergence rates, requiring significantly larger sample sizes to obtain satisfactory approximations; they are also strongly affected by the dimensionality of the data and the selection of the smoothing parameter is a crucial step. In contrast, they only require some kind of smoothness assumption on the target density.

Neural Networks (Hertz et. al 1991) are computational models trainable from empirical data that have been proposed to solve more complex situations. Their intrinsic parallel architecture is especially efficient in the inference stage. One of the most widely used neural models is the Multilayer Perceptron, a universal function approximator (Hornik et al. 1989) that breaks the limitations of linear decision functions. The Backpropagation learning algorithm (Rumelhart et al. 1986) can, in principle, adjust the network weights to implement arbitrary mappings, and the network outputs show desirable probabilistic properties (Wan 1990, Rojas 1996). There are also unsupervised networks for probability density function approximation (Kohonen 1989). However, neural models usually contain a large number of adjustable parameters, which is not convenient for generalization and, frequently, long times are required for training in relatively easy tasks. The input / output role of attributes cannot be changed in runtime and missing and uncertain values are poorly supported.

Bayesian Networks, based in the concept of conditional independence, are among the most relevant probabilistic inference technologies (Pearl 1988, Heckerman & Wellman 1995). The joint density of the variables is modeled by a directed graph which explicitly represents dependence statements. A wide range of inferences can be performed under this framework (Chang & Fung 1995, Lauritzen & Spiegelhalter 1988) and there are significant results on inductive learning of network structures (Bouckaert 1994, Cooper & Herskovits 1992, Valiveti & Oomen 1992). This approach is adequate when there is a large number of variables showing explicit dependences and simple cause-effect relations. Nevertheless, solving arbitrary queries is NP-Complete, automatic learning algorithms are time consuming and the allowed dependences between variables are relatively simple. In an attempt to mitigate some of the above drawbacks, we have developed a general and efficient inference and learning framework based on the following considerations. It is well known
(Titterington et al. 1985, McLachlan & Basford 1988, Dalal & Hall 1983, Bernardo & Smith 1994, Xu & Jordan 1996) that any reasonable probability density function p(z) can be approximated up to the desired degree of accuracy by a finite mixture of simple components Ci, i = 1..l:
p Cip Ci i
(z) ≅ ∑ P{ } (z| ) (1)
The superposition of simple densities is extensively used to approximate arbitrary data dependences (Fig. 2). Maximum Likelihood estimators of the mixture parameters can be efficiently obtained from samples by the Expectation Maximization (EM) algorithm (Dempster, Laird & Rubin 1977, Redner & Walker 1984) (see Section 4).