This means that the sum of products in equation (9) is "complete", i.e., it includes all the elements in the N-fold cartesian product of attributes:

$$p({\bf S}|{\bf z})=\prod_{i}\sum_{r}\pi_{r}^{j}\ {\mathcal N}(z^{j},s_{r}^{j},\varepsilon_{r}^{j})$$
j r
where r j Π jπr = π . This factored likelihood function can be considered also as a 1-component mixture (with R=1 in (9) and s s j j
≡ 1
) where the marginal observation models are allowed to be mixtures of generalized normals: p s z z s j j r r j j r j r j
( | ) ( , , ) = Σ 'π ' ' ' T ε . In this case we can even think of "function valued" attributes z ≅ [ f (z ),..., f (z )]
1 1 n n, where f z p s z j j j j
( ) ≡ ( | ) models the range and relative likelihood of the values of z j. Loosely speaking, attributes with concentrated f z j j
( ) may be considered as inputs, and attributes with high dispersion play the role of outputs.

Since y is conditionally independent of s xgiven Ci
, the posterior can be obtained from the expansion:

$$p(\mathbf{y}|\mathbf{S})=\sum_{i}p(\mathbf{y}|\mathbf{S},C_{i})\;\mbox{P}\{C_{i}|\mathbf{S}\}=\sum_{i}p(\mathbf{y}|C_{i},\mathbf{s}^{\,\mathbf{y}})\;\mbox{P}\{C_{i}|\mathbf{S}\}\tag{19}$$

The interpretation of (19) is straightforward. The effect of s x over y = {z d} must be computed through x = {z o} and the components Ci
. Then, a simple Bayesian update of p( y|s )
xas a new prior is made using s y(see Fig. 15).

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

