
| πr                                                                   | sr    | r    |       |           |      |      |      |
|----------------------------------------------------------------------|-------|------|-------|-----------|------|------|------|
| 1                                                                    | 1     |      |       |           |      |      |      |
| ,ε                                                                   | sr    | r    |       |           |      |      |      |
| 2                                                                    | 2     |      |       |           |      |      |      |
| ,ε                                                                   | sr    | r    |       |           |      |      |      |
| 3                                                                    | 3     |      |       |           |      |      |      |
| , ε                                                                  | sr    | r    |       |           |      |      |      |
| 4                                                                    | 4     |      |       |           |      |      |      |
| ,ε                                                                   | sr    | r    |       |           |      |      |      |
| 5                                                                    | 5     |      |       |           |      |      |      |
| ,ε                                                                   | sr    | r    |       |           |      |      |      |
| 6                                                                    | 6     |      |       |           |      |      |      |
| ,ε                                                                   | sr    | r    |       |           |      |      |      |
| 7                                                                    | 7     |      |       |           |      |      |      |
| ,ε                                                                   |       |      |       |           |      |      |      |
| .30                                                                  | a, εa | -, ∞ | b, εb | -, ∞      | -, ∞ | -, ∞ | -, ∞ |
| .63                                                                  | c, εc | -, ∞ | -, ∞  | triang, 0 | -, ∞ | -, ∞ | d,εd |
| .63                                                                  | c, εc | -, ∞ | -, ∞  | triang, 0 | -, ∞ | -, ∞ | e, 0 |
| .07                                                                  | c, εc | -, ∞ | -, ∞  | circ, 0   | -, ∞ | -, ∞ | d,εd |
| .07                                                                  | c, εc | -, ∞ | -, ∞  | circ, 0   | -, ∞ | -, ∞ | e, 0 |
| Table 2. Parameters of the Uncertain Information Model in Example 8. |       |      |       |           |      |      |      |

## 3.3 The Joint Model-Observation Density

The generic dependence structure in Fig. 6 is implemented by the MFGN framework as shown in Fig. 8. The upper section of the figure is the model of nature, obtained in a previous learning stage and used for inference without further changes. Dependences among attributes are conducted through an intermediary hidden or latent component Ci. The lower section represents the available uncertain information, measurement model or query structure associated to each particular inference operation.

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

The joint density of the relevant variables becomes:

$$p(C_{i},z,s_{r},S)=\mbox{P}\{S|s_{r}\}\ p(s_{r}|z)\ p(z|C_{i})\ \mbox{P}\{C_{i}\}$$ $$=\mbox{P}\{C_{i}\}\ \mbox{P}\{S|s_{r}\}\prod_{j}p(s_{r}^{j}|z^{j})\ p(z^{j}|C_{i})$$ $$=\ P_{i}\ \pi_{r}\prod_{j}\mathcal{N}(z^{j},s_{r}^{j},\epsilon_{r}^{j})\ \mathcal{N}(z^{j},\mu_{i}^{j},\sigma_{i}^{j})\tag{10}$$
