$\beta_{i,r}^{j}=\mathcal{N}(s_{r}^{j},\mu_{i}^{j}-\eta_{r}^{j},\sqrt{(\sigma_{i}^{j})^{2}+(\epsilon_{r}^{j})^{2}})$  put $z^{j}=s_{r}^{j}\pm2\epsilon_{r}^{j}$ is equivalent to the effet

$=\;\boxed{\phantom{\frac{1}{1}}}$
$$\overline{{\gamma}}$$
The effect of a noisy input z s
≅ ± 2ε is equivalent to the effect of an exact input z s

$${\mathrm{an~exact~input~}}z^{j}=s_{r}^{j}$$  In aoxtoxtoxtr; anr</r>a, the r. 
in a mixture with components of larger variance: σ σ ε i j i j r → j
( ) + ( )
2 2. Uncertainty spreads the effect of the observation, increasing the contribution of distant components.

Example 12: Fig. 13.a shows a simple two-attribute domain approximated by a 3-component mixture. We are interested in the marginal density of attribute x given different degrees of uncertainty ε in the input attribute y ≈ .4 ± 2ε, modeled by p s y s y y
( | ) = T( ,.4,ε) . If ε = 0 we have the sharpest density (A) in Fig. 13.b, providing x≈–.4±.5. If ε = .25 we obtain density
(B) and x≈–.3±.7. Finally, if ε = .5 we obtain density (C) and x≈–.2±.8. Obviously, as the uncertainty in y increases, so does the uncertainty in x. The expected value of x moves towards more distant components, which become more likely as the probability distribution of y expands. In this situation an interesting effect appears: the mode of the marginal density does not change at the same rate than the mean. Uncertainty in y skews p(x). This effect suggests that the optimum estimators for different loss functions are not equally robust against uncertainty.

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

For the output role, ψi r j j z ,
( ) becomes the original marginal, modified in location and dispersion towards sr jaccording to the factor γ = (σ ) /[(σ ) + (ε ) ]
i j i j r 2 2 j 2, which quantifies the relative importance of the observation:

$$\Psi_{i,r}^{j}(z^{j})=\mathcal{N}\Big(z^{j},\gamma(s_{r}^{j}-\eta_{r}^{j})+(1-\gamma)\mu_{i}^{j},\ \gamma^{1/2}\varepsilon_{r}^{j}\Big)$$

Missing data. When there is no information about the j-th attribute, s z r j j
= { = ?}, the observation can be modeled by p s z r j j
( | ) = *constant* or, equivalently, p s z s a b r j j r j
( | ) = T( , , ) with a arbitrary and b → ∞. All the components contribute with the same weight:
βi r j j i p z anything C *constant*
, = ( = | ) ∝ ≡ 1 If the target is missing the ψi r j j z ,
( ) reduce to the original marginal components: