Now we will derive an alternative expression for eq. (10) which is more convenient for computing the marginal densities of any desired variable. Using the following relation:

p sr
z p z C p z s C p z s C p s C
j j j
i
j
r
j
i
j
$$p(s_{r}\mid s_{r},C_{i})\ p(s_{r}\mid C_{i})$$
j
j
( | ) ( | ) = ( , | ) = ( | , ) ( | )
and properties (2) and (3), we define the "dual" densities of the model:

$$\beta_{i,r}^{j}\equiv p(s_{r}^{j}|C_{i})=\int_{z^{j}}p(s_{r}^{j}|z^{j})\ p(z^{j}|C_{i})\ d z^{j}=\mathcal{N}(s_{r}^{j},\mu_{i}^{j},\rho_{i,r}^{j})$$
ψi rν λ
$$\Psi_{i,r}^{j}(z^{j})\equiv p(z^{j}|s_{r}^{j},C_{i})=$$
$$\frac{p(s_{r}^{j}|z^{j})}{p(s_{r}^{j}|C_{i})}\,p(z^{j}|C_{i})=\mathcal{N}(z^{j},\mathbf{v}_{i,r}^{j},\lambda_{i,r}^{j})$$
p z C z , , ,
≡ = ( | ) = T( , , ) (12)
where the parameters $\rho_{i,r}^{j}$, $\nu_{i,r}^{j}$ and $\lambda_{i,r}^{j}$ are given by:  $$\rho_{i,r}^{j}\equiv\sqrt{\left(\sigma_{i}^{j}\right)^{2}+\left(\epsilon_{r}^{j}\right)^{2}}$$  $$\nu_{i,r}^{j}\equiv\frac{\left(\sigma_{i}^{j}\right)^{2}s_{r}^{j}+\left(\epsilon_{r}^{j}\right)^{2}\mu_{i}^{j}}{\left(\rho_{i,r}^{j}\right)^{2}}$$  $$\lambda_{i,r}^{j}\equiv\frac{\sigma_{i}^{j}\epsilon_{r}^{j}}{\rho_{i,r}^{j}}$$
$$(11)$$
$$(12)$$
$$(13)$$

βi r j
,is the likelihood of the r-th elementary observation sr j of the j-th attribute z jin each component Ci and ψi r j j z ,
( ) is the effect of the r-th elementary information sr jabout the j-th attribute z j over the marginal component p(z j|Ci
) in each component Ci
. Using the above notation, the MFGN model structure can be conveniently written as:

$$p(C_{i},z,s_{r},S)=P_{i}\,\pi_{r}\prod_{j}\beta_{i,r}^{j}\,\,\Psi_{i,r}^{j}(z^{j})$$
z s S = π ∏β ψ (13)

## 3.4 The Posterior Density

In the inference process the available information is combined with the model of the domain to update our knowledge about a particular object. Given a new piece of information S we must compute the posterior distribution p( y|S) of the desired target attributes y ⊆ z. Then, estimators y(S) ≅ y can be obtained from p( y|S) to minimize any suitable average loss function. This is efficiently supported under the MFGN framework regardless of the complexity of the domain p(z)
and the structure of the available information S = {π s } r r
.

The attributes are partitioned into two subvectors z = (x, y), where y = { z d} are the desired target attributes and x = { z o} are the rest of attributes. Accordingly, each component sr of the available information S is partitioned as s s s x y r = r r
( , ) . The information about the target attributes y in the r-th observation, independent from the model p(z), is denoted by s y r(often y is just missing and there are no such pieces of information) and s x rrepresents the information about the rest of attributes x. Using this convention we can write: