In realistic situations the variables are statistically dependent. In general, the joint density cannot be factorized and the required marginal densities may be hard to compute. However, interesting consequences arise if the joint density is expressed as a finite mixture of factorized (with independent variables) components C1, C2, ..., Cl
:

$$p(z)=p(z^{1},...,z^{n})=\sum_{i}\ {\rm P}\{C_{i}\}\ p(z|C_{i})=\sum_{i}\ {\rm P}\{C_{i}\}\prod_{j}p(z^{j}|C_{i})\tag{5}$$

This structure is convenient for inference purposes. In particular, in terms of the desired partition of z = (x, y):
$$p(z)=\,p(x,y)=\sum_{i}\,\mathrm{P}\{C_{i}\}\;p(x|C_{i})\;p(y|C_{i})$$
i so the marginal densities are mixtures of the marginal components:

e mixtures of the marginal components:  $$p(\mathbf{x})=\int_{\mathbf{y}}p(\mathbf{x},\mathbf{y})d\mathbf{y}=\sum_{i}\mathbf{P}\{C_{i}\}\ p(\mathbf{x}|C_{i})$$  $$p(\mathbf{y})=\sum_{i}\mathbf{P}\{C_{i}\}\ p(\mathbf{y}|C_{i})$$
i
and the desired conditional densities are also mixtures of the marginal components:

$$p(\mathbf{y}|\mathbf{x})=\sum_{i}\alpha_{i}(\mathbf{x})\ p(\mathbf{y}|C_{i})$$
( y|x) = ∑α (x) ( y| ) (6)
where the weights αi
(x) are the probabilities that the observed x has been generated by each component Ci
:

$$\alpha_{i}(x)={\frac{\mathrm{P}\{C_{i}\}p(x|C_{i})}{\sum_{j}\mathrm{P}\{C_{j}\}p(x|C_{j})}}=\mathrm{P}\{C_{i}|x\}$$
$$(6)$$
j
The p.d.f. approximation capabilities of mixture models with factorized components remain

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

unchanged, at the cost of a possibly higher number of components to obtain the desired degree of accuracy, avoiding "artifacts" (see Fig. 5). Section 5.2 discusses the implications of factorization in relation with alternative model structures.