mated (see Section 4) by a 6–component mixture (the error rate classifying U in the validation

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

set without rejection was 2.67%). Fig. 14 shows two projections of the 150 examples and the location of the mixture components learned from the training subset. The parameters of the mixture are shown in Table 3.

| i                                                        | Pi   | µi x   | σi x   | µi y   | σi y   | µi z   | σi z   | µi w   | σi w   | P{U1|Ci}   | P{U2|Ci}   | P{U3|Ci}   |
|----------------------------------------------------------|------|--------|--------|--------|--------|--------|--------|--------|--------|------------|------------|------------|
| 1                                                        | 0.15 | 7.13   | 0.48   | 3.12   | 0.34   | 6.17   | 0.45   | 2.18   | 0.20   | 0          | 0          | 1          |
| 2                                                        | 0.13 | 5.48   | 0.41   | 2.50   | 0.28   | 3.87   | 0.32   | 1.20   | 0.21   | 0          | 0.93       | 0.07       |
| 3                                                        | 0.21 | 6.29   | 0.39   | 2.93   | 0.27   | 4.59   | 0.20   | 1.45   | 0.14   | 0          | 1          | 0.00       |
| 4                                                        | 0.18 | 4.75   | 0.23   | 3.25   | 0.23   | 1.42   | 0.21   | 0.19   | 0.05   | 1          | 0          | 0          |
| 5                                                        | 0.15 | 5.36   | 0.26   | 3.76   | 0.29   | 1.51   | 0.16   | 0.32   | 0.10   | 1          | 0          | 0          |
| 6                                                        | 0.17 | 6.16   | 0.42   | 2.77   | 0.28   | 5.22   | 0.30   | 1.94   | 0.23   | 0          | 0.00       | 1          |
| Table 3. Parameters of the Iris Data Joint Density Model |      |        |        |        |        |        |        |        |        |            |            |            |

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

Table 4 shows the results of the inference process in the following illustrative situations: Case 1: Attribute z is known: S = {z = 5}.

Case 2: Attributes x and U are known: S = {(x = 5.5) AND (U=U2)}.

Case 3: Attribute x is uncertain: S = {x ≅ 7±1}.

Case 4: Attributes x and w are uncertain: S = {(x ≅ 7±1) AND (w ≅ 1±0.5)}. Note that uncertainty decreases when more information is supplied (compare with Case 3).

Case 5: Structured query expressed in terms of logical connectives over uncertain elementary events: S = {[(z ≅ 1±3) OR (z ≅ 7±3)] AND [(U = U1) OR (U = U2)]}.