
$$p(z,S)=p(x,y,S)=\sum_{i,r}P_{i}\ \pi_{r}\ \beta_{i,r}\ \Psi_{i,r}(x)\ \Psi_{i,r}(y)$$

Where βi,r is the likelihood of the r-th conjunction sr of S in component Ci
:

$$\beta_{i,r}\equiv\prod_{i}\beta_{i,r}^{j}$$
$$(14)$$

, ≡ ∏ ,(14)
and the terms ψi r j j z ,
( ) are grouped according to the partition of z = (*x, y*):

$$\Psi_{i,r}(x)\equiv\prod_{a}\psi_{i,r}^{o}(z^{o})\qquad\qquad\Psi_{i,r}(y)\equiv\prod_{d}\psi_{i,r}^{d}(z^{d})$$
o
d
The desired posterior p(y|S) = p(y,S) / p(S) can be computed from the joint p(z,S) by marginalization: along x to obtain p(y,S) and along all z to obtain p(S). Note that each univariate marginalization of p(z,S) along attribute z jeliminates all the terms ψi r j j z ,
( ) in the sum (13):

$p(y,S)=\int_{X}p(x,y,S)\,dx=\sum_{i,r}P_{i}\,\pi_{r}\,\beta_{i,r}\,\Psi_{i,r}(y)$  $p(S)=\int_{Z}p(z,S)\,dz=\sum_{i,r}P_{i}\,\pi_{r}\,\beta_{i,r}$
Therefore, the posterior density can be compactly written as:

$$p(\mathbf{y}|\mathbf{S})=\sum_{i}\alpha_{i,r}\Psi_{i,r}(\mathbf{y})$$
$\sigma$
i r
,
y S = ∑α Ψ y (15)
where αi,ris the probability that the object z has been generated by component Ciand the elementary information sr is true, given the total information S:

$$\alpha_{i,r}\equiv\mathrm{P}\{C_{i},s_{r}|S\}=\frac{P_{i}\ \pi_{r}\ \beta_{i,r}}{\sum_{k,l}P_{k}\ \pi_{l}\ \beta_{k,l}}\tag{1}$$

$$(15)$$
$$(16)$$
and $\Psi_{i,r}(\mathbf{y})=p(\mathbf{y}|\mathbf{s}_{r}^{\dagger},C_{i})$ is the marginal density $p(\mathbf{y}|C_{i})$ of the desired attributes in the $i$-th
component, modified by the contribution of all the associated s y r
. Since p r Ci
( y|s , )
y=
p r Ci
( y|s , ) , the expression (16) also follows from the expansion:

$$p(\mathbf{y}|\mathbf{S})=\sum_{i,r}p(\mathbf{y}|\mathbf{s}_{r},\mathbf{C}_{i})\operatorname{P}\{C_{i},\mathbf{s}_{r}|\mathbf{S}\}$$

In summary, when the joint density and the likelihood function are approximated by mixture models with the proposed structure, the computation of conditional densities given events of arbitrary "geometry" is notably simplified. Factorized components reduce multidimensional integration to simple combination of univariate integrals and conjugate families avoid numeric integration. This property is illustrated in Fig. 9.