

![0_image_0.png](0_image_0.png) 

## 3.5 Expressions For The Estimators

Approximations to the optimum estimators can be easily obtained by taking advantage of the mathematically convenient structure of the posterior density. Under the MFGN framework, the conditional expected value of any function g(y) becomes a linear combination of constants:
E{g g p d Y
( y)|S} = ( y) ( y|S) y ∫ =

$$=\sum_{i,r}\,\alpha_{i,r}\,\int_{\gamma}g(y)\,\Psi_{i,r}\,(y)\,dy\,=\,\sum_{i,r}\,\alpha_{i,r}\,\,{\rm E}_{i,r}\,\{g(y)\}\tag{17}$$

where E E{ i,r g g r Ci
{ ( y)} ( y)|s , }
$\mathbf{r}\cdot\mathbf{G}\lambda\cdot\mathbf{i}=\lambda\mathbf{i}$
≡ is the expected value of g(y) in the i-th component
2modified 3 by the r-th observation s y r
:

$$\operatorname{E}_{i,r}\{g(y)\}\equiv\int_{Y}g(y)\prod_{d}{\mathcal{N}}(z^{d},\mathbf{v}_{i,r}^{d},\lambda_{i,r}^{d})\,d y$$

We can now analytically compute the desired optimum estimators. For instance, the MSE estimator for a single continuous attribute y = z
drequires the mean values Ei rdi r
d
z , ,
{ } = ν :
$${\hat{y}}_{M S E}\left(S\right)=\operatorname{E}\{y|S\}=\sum_{i}\ \alpha_{i,r}\ \mathbf{v}_{i,r}^{d}$$
,
i r From our explicit expression for p(y|S) we can also compute the conditional cost:

$$e_{M S E}^{2}(S)=\mathrm{E}\Bigl\{\bigl(y-\hat{y}_{M S E}(S)\bigr)^{2}|S\Bigr\}=\mathrm{E}\{y^{2}|S\}-\hat{y}_{M S E}^{2}(S)=$$
$$=\sum_{i,r}\,\alpha_{i,r}\,\left[\left({\bf v}_{i,r}^{d}\right)^{2}+\left(\lambda_{i,r}^{d}\right)^{2}\right]\,-\left(\sum_{i,r}\,\alpha_{i,r}\,\,{\bf v}_{i,r}^{d}\right)^{2}$$
