
| CASE                                                 | X          | y         | z         | w       | U               |                 |
|------------------------------------------------------|------------|-----------|-----------|---------|-----------------|-----------------|
| 1                                                    | INPUT      | ?         | ?         | 5.0     | ?               | ?               |
| OUTPUT                                               | 6.2±0.9    | 2.8±0.6   | 5.0       | 1.8±0.6 | U2: 22% U3: 78% |                 |
| 2                                                    | INPUT      | 5.5       | ?         | ?       | ?               | U2: 100%        |
| OUTPUT                                               | 5.5        | 2.6±0.6   | 4.0±0.8   | 1.3±0.4 | U2: 100%        |                 |
| 3                                                    | INPUT      | 7±1       | ?         | ?       | ?               | ?               |
| OUTPUT                                               | 6.7±0.9    | 3.0±0.7   | 5.3±1.8   | 1.8±0.8 | U2: 36% U3: 63% |                 |
| 4                                                    | INPUT      | 7±1       | ?         | ?       | 1±0.5           | ?               |
| OUTPUT                                               | 6.5±0.7    | 2.9±0.6   | 4.5±0.8   | 1.3±0.3 | U2: 95% U3: 5%  |                 |
| 5                                                    | INPUT      | ?         | ?         | 1±3     | ?               | U1: 50% U2: 50% |
| OR                                                   | ?          | ?         | 7±3       | ?       | U1: 50% U2: 50% |                 |
| OUTPUT                                               | 5.3±1.2    | 3.3±0.9   | 2±3       | 0.5±1   | U1: 75% U2: 25% |                 |
| (approx.                                             | (unimodal) | (bimodal) | (bimodal) |         |                 |                 |
| unimodal)                                            |            |           |           |         |                 |                 |
| Table 4. Some Inference Results Over the IRIS Domain |            |           |           |         |                 |                 |

The consistency of the results can be visually checked in Fig. 14. Finally, Table 5 shows the elementary likelihoods j βi,r of Case 5, illustrating the essence of the method.

i βi

x

,1 βi

y

,1 βi

z

,1 βi

w

,1 βi

U

,1 βi,1 βi

x

,2 βi

y

,2 βi

z

,2 βi

w

,2 βi

U

,2 βi,2

1 1 1 .001 1 0 0 1 1 **.221** 1 0 0

2 1 1 .045 1 .47 .02 1 1 .032 1 .47 .02

3 1 1 .016 1 .50 .01 1 1 .074 1 .50 .04 4 1 1 .254 1 .50 .13 1 1 3E-4 1 .50 .00

5 1 1 .250 1 .50 .13 1 1 4E-4 1 .50 .00

6 1 1 .006 1 0 0 1 1 **.132** 1 0 0

Table 5. Elementary likelihoods in Case 5 from Table 4.

## 3.8 Independent Measurements

One of the key features of the MFGN framework is the ability to infer over arbitrary relational knowledge about the attributes, in the form of a likelihood function adequately approximated by a mixture model with the structure of eq. (9). For instance, we could answer questions as: "what happens to z dwhen z itends to be less than z j?" (i.e., when p(S|z) is high in the region z z i j
− < 0 ).

However, there are situations where the observations over each single attribute z jare statistically independent: we have information about attributes (e.g. z iis around a and z jis around b) but not about attribute relations. We will pay attention to this particular case because it illustrates the role of the main MFGN framework elements. Furthermore, many practical applications can be satisfactorily solved under the assumption of independent measurements or judgments. In this case, the likelihood of the available information can be expressed as the conjunction of n "marginal" observations s jabout z j:

$$p(S|z)=\prod_{i}p(s^{j}|z^{j})$$
(S|z) = ∏ ( | ) (18)
$$(18)$$
