

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

Figure 4: The plot of the upper bound on the partial derivative ∂Pr (y | e)/∂τx|u**, as given**
in Theorem 3.1, against Pr (x | u**) and** Pr (y | e).

Theorem 3.1 If X **is a binary variable in a belief network, then:**4

$$\left|{\frac{\partial P r(y\mid\mathbf{e})}{\partial\tau_{x|\mathbf{u}}}}\right|\leq{\frac{P r(y\mid\mathbf{e})(1-P r(y\mid\mathbf{e}))}{P r(x\mid\mathbf{u})(1-P r(x\mid\mathbf{u}))}}$$

The bound in Theorem 3.1 is tight, and we will show later an example for which the derivative assumes the above bound exactly. The main point to note about this bound is that it is independent of any given belief network.5 The plot of this bound against Pr (x | u) and Pr (y | e**) is shown in Figure 4. A number**
of observations are in order about this plot:

- For extreme values of Pr (x | u**), the bound approaches infinity, and thus a small**
absolute change in the meta parameter τx|u **can have a big impact on the query**
Pr (y | e).

- **On the other hand, the bound approaches 0 for extreme values of the query** Pr (y | e).

Therefore, a small absolute change in the meta parameter τx|u **will have a small effect**
on the absolute change in the query.
One of the implications of this result is that if we have a belief network where queries of interest Pr (y | e**) have extreme values, then such queries will be robust against small**
changes in network parameters. This of course assumes that robustness is understood to