urement or a more subjective judgment about the location of z in the attribute space. All we need to update our knowledge about z, in the form of the posterior p(z|S), is the relative likelihood function p(S|z) of the observed S. In general, p(S|z) can be any nonnegative multivariable function fS(z) over the domain. In the objective case, statistical studies of the measurement process can be used to determine the likelihood function. In the subjective case, it may be obtained by standard distribution elicitation techniques (Berger 1985). In either case, under the MFGN framework, the likelihood function of the available information to be used in the inference process will be approximated, up to the desired degree of accuracy, by a sum of products of generalized normals:

$$p(S/z)=\sum_{r}\mathrm{P}\{S|s_{r}\}p(s_{r}|z)=\sum_{r}\mathrm{P}\{S|s_{r}\}\prod_{j}p(s_{r}^{j}|z^{j})$$
$$=\sum_{r}\pi_{r}\,\prod_{j}{\mathcal{N}}(z^{j},s_{r}^{j},\epsilon_{r}^{j})$$
$$(9)$$

∑ ∏T(z ,s , ) (9)
Without loss of generality, the available knowledge is structured as a weighted disjunction S =
{π1s1 OR π2s2 ... OR π R R
s } of conjunctions sr = { sr 1AND sr 2 ... AND sr n) of elementary uncertain

```
observations in the form of generalized normal likelihoods T(z ,s , )
                                                                 j
                                                                    r
                                                                    j
                                                                        r
                                                                        j
                                                                      ε centered at sr
                                                                                        j with

```

uncertainty εr j. The measurement process can be interpreted as the result of R (objective or subjective) sensors sr
, providing conditionally independent information p s z r j j
( | ) about the attributes
(each sr j only depends on z j) with relative strength πr
. Note that any complex uncertain information about an instance z, expressed as a nested combination of elementary uncertain beliefs sr j about z j using "probabilistic connectives", can be ultimately expressed by structure (9) (OR translates to addition, AND translates to product and the product of two generalized normals over the same attribute becomes a single, weighted normal).

Example 8: Consider the hypothetical computer vision domain in Example 1. Assume that the information about an object z is the following: "AREA is around a and DISTANCE is around b or, more likely, SHAPE is surely *triangular* or else *circular* and AREA is around c and ANGLE is around d or equal to e". This structured piece of information can be formalized as:

 p(S|z) = .3 [T(z
$$p({\bf S}|{\bf z})=.3\,[{\cal N}(z^{1},a,\,\varepsilon_{a}\,)\,{\cal N}(z^{3},b,\,\varepsilon_{b}\,)]$$
 + .7 [ (.9T(z
$$+.7\;[\;(.9{\mathcal{N}}(z^{4},\!r i a n g)+.1{\mathcal{N}}(z^{4},\!c i r c))\;{\mathcal{N}}(z^{1},c,\;\epsilon_{c})\;({\mathcal{N}}(z^{7},d,\;\epsilon_{d})+{\mathcal{N}}(z^{7},\!e))\;]\;,$$
which, expanded, becomes the mixture of 5 factorized components operationally represented by the parameters shown in Table 2.

In a simpler situation, the available information about z could be a conjunction of uncertain attributes similar to {Color = red 0.8 green 0.2} and {Area = 3 ± .5} and {Shape = rectangular 0.6 circular 0.3 triangular 0.1}. The likelihood of Shape values can be obtained from the output of a simple pattern classifier (e.g. K-N-nearest neighbors) over moment invariants, while attributes as Color and Area are directly extracted from the image. In this case we could be interested in the distribution of values for other attributes as Texture and ObjectType. Alternatively, we could start from {ObjectType = door 0.6 window 0.4} and {Texture = rough} in order to determine the probabilities of Color and Angle values for selecting a promising search region.