
## 5. Discussion And Experimental Results 5.1 Advantages Of Joint Models

Most inductive inference methods compute a direct approximation of the conditional densities of interest, or even obtain empirical decision rules without explicit models of the underlying conditional densities. In these cases, both the model and the learning stage depend on the selected input /
output role of the variables. In contrast, we have presented an inference and learning method based on an approximation to the joint probability density function of the attributes by a convenient parametric family (a special mixture model). The MFGN framework works as a pattern completion machine operating over possibly uncertain information. For example, given a pattern classification problem, the same learning stage suffices for predicting class labels from feature vectors and for estimating the value of missing features from the observed information in incomplete patterns. The joint density approach finds the regions occupied by the training examples in the whole attribute space. The attribute dependences are captured at a higher abstraction level than the one provided by strictly empirical rules for pre-established target variables. This property is extremely useful in many situations, as shown in the following examples.

Example 15: *Hints* can be provided for inference over multivalued relations. Given the data

![0_image_0.png](0_image_0.png) set and model from Example 10, assume that we are interested in the value of x for y = 0. We obtain the bimodal marginal density shown in Fig. 19.a and the corresponding estimator x ≅ 0.2 ± 1.4 which is, in some sense, meaningless. However, if we specify the *branch* of interest of the model, inferring from y = 0 AND x ≅ -1±1 (i.e., "x is small"), we obtain the unimodal marginal density in Fig. 19.b and the reasonable estimator x ≅ –0.8±0.5.

Example 16: *Image Processing*. The advantages of a joint model supporting inferences from partial information on both inputs and outputs can be illustrated in the following application with natural data (see Fig. 20). The image in Fig. 20.a is characterized by a 5-attribute density (x, y, *R, G*, B) describing position and color of the pixels. A random sample of 5000 pixels was used to build a 100-component mixture model. We are interested in the location of certain ob-