
## 3. The Mfgn Framework

The previous concepts will be integrated in a general probabilistic inference framework that we call MFGN (*Mixtures of Factorized Generalized Normals*). Fig. 6 shows the abstract dependence relations among attributes in a generic domain (upper section of the figure) and between the attributes and the observed information (lower section). In the MFGN framework, both relations are modeled by finite mixtures of products of generalized normals. The key idea is using factorization to cope with multivariate domains and heterogeneous attribute vectors, and conjugate densities to efficiently perform inferences given arbitrary uncertain information. In this section, we will derive the main inference expressions. The learning stage will be described in Section 4.

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

## 3.1 Modeling Attribute Dependences In The Domain

In the MFGN framework the attribute dependencies in the domain are modeled by a joint density in the form of a finite mixture of factored components, as in expression (5), where the component marginals p z C z j i j i j i j
( | ) ≡ T( ,µ ,σ ) are generalized normals:

$$p(z)=\sum_{i}P_{i}\prod_{j}{\cal N}(z^{j},\mu_{i}^{j},\sigma_{i}^{j})\qquad i=1..l,\;j=1..n,\tag{7}$$

If desired, the terms associated to the pure symbolic attributes z j(with all the σi j= 0) can be collected in such a way that the component marginals are expressed as mixtures of impulses:

$$p(z^{j}|C_{i})\equiv\sum_{\omega}t_{i,\omega}^{j}{\mathcal{N}}(z^{j},\omega)$$
$$({\boldsymbol{\delta}})$$
$\vec{l}=e\hat{r}$. 
T ω (8)
where tiz C
j j
, i
{ | } ω ≡ P = ω is the probability that z jtakes its ω-th value in component Ci
. This manipulation reduces the number l of global components in the mixture. The adjustable parameters of the model are the proportions Pi = P Ci
{ } and the mean value µi jand dispersion σi j of the j-th attribute in the i-th component (or, for the symbolic attributes, the probabilities j i t,ω
). While the structure (8) will be explicitly used for symbolic attributes in applications and illustrative examples, most of the mathematical derivations will be made over the concise expression (7).

pt $C_{\mathrm{m}}$ Th. 