When all variables are continuous, the MFGN architecture reduces to a mixture of gaussians with diagonal covariance matrices. The proposed factorized structure extends the properties of diagonal covariance matrices to heterogeneous attribute vectors. We are interested in joint models, which support inferences from partial information about any subset of variables. Note that there is not an easy way to define a measure of statistical depencence between symbolic and continuous attributes, to be used as a parameter of some probability density function 1. The required "hetereogeneous" dependence model can be conveniently captured by superposition of simple factorized
(with independent variables) densities.

Example 7: Figure 7 shows an illustrative 3-attribute data set (x and y are continuous and z is

![0_image_0.png](0_image_0.png)

symbolic) and the components of the MFGN approximation obtained by the EM algorithm
(see Section 4) for their joint density. The parameters of the mixture are shown in Table 1.

Note that, because of the overlapped structure of the data, some components (5 and 6) are "assigned" to both values of the symbolic attribute z.

![0_image_1.png](0_image_1.png) 

## 3.2 Modeling Arbitrary Information About Instances

The available information about a particular instance z is denoted by S. Following the likelihood principle, we are not concerned with the true nature of S, whether it is some kind of physical meas-