
## Ψi R Μ Σ J J J I J I J J I , (Z ) = T(Z , , ) = P(Z |C )

Arbitrary uncertainty. In general, any unidimensional relative likelihood function can be approximated by a mixture of generalized normals, as shown in Example 6, where βi r j
,and ψi r j j z ,
( )
are given respectively by eqs. (11) and (12). Intervals. Some useful functions cannot be accurately approximated by a small number of normal components. A typical example is the indicator function of an interval, used to model an uncertain observation where all the values are equally likely: s {z (a,b)} j j r = ∈ . If z jis only considered as input, we can use the shortcut ( ) ( ) , F b F a j i j i j βi r = − , where F z i j j
( ) is the cumulative distribution of the normal marginal component p z C
j i
( | ) . Unfortunately, the expression for ψi r j j z ,
( ),
required for z jconsidered as output, may not be so useful for computing certain optimum estimators. ψi r j j z ,
( ) is the restriction of *p(z* j|Ci) to the interval (a,b) and normalized by βi r j
,
.

Disjunction and conjunction of events. Finally, standard probability rules are used to build structured information from simple observations: if from subjective judgments or objective evidence we ascribe relative degrees of credibility θr jto several observations sr jabout z j, the overall likelihood becomes βi θ β j r r j i r j
= Σ ,
. In particular, if s z z j j j
= { = ω = ω } 1 OR 2and the two possibilities are equiprobable then βi ω ω j i i = p( |C ) + p( |C ) 1 2
. Analogously, conjunctions of events translate to multiplication of likelihood functions.

## 3.7 Summary Of The Inference Procedure

Once the domain p(z) has been adequately modeled in the learning process (as explained in Section 4), the system enters the inference stage over new, partially specified objects. From the parameters of the domain p(z) ( Pi i j i j
, µ and σ ) and the parameters of the model of the observation p(S|z)
( π ε r r j r j
, s and ), we must obtain the parameters βi r ν λ j i r d i r d
, , ,
, and of the desired marginal posterior densities and estimators. The inference procedure comprises the following steps:
 Compute the elementary likelihoods βi r j
,
, using eq. (11).

 Obtain the product βi,r for each conjunction sr and component Ci
, using eq. (14).

 Normalize Pi πr βi,r to obtain the coefficients αi,r of the posterior, using eq. (16).

 Choose the desired target attributes y = { z d} and compute the parameters νi r d
,
, and λi r d
,
 of the modified component marginal densities ψi r d d z ,
( ) using eq. (12).

 Report the joint posterior density of y. Show graphs of the posterior marginal densities of the desired attributes z d using eq. (15). Provide optimum (point, interval, etc.) estimators using eq. (17).
Example 13: *Iris Data*. The inference procedure is illustrated over the well known Iris benchmark: 150 objects represented by four numeric features (x, y, z and w) and one symbolic category U ∈ {U1 (*setosa*), U2 (*versicolor*), U3 (*virginica*)}. The whole data set was divided into two disjoints subsets for training and validation. The joint density can be satisfactorily approxi-