From Figure 10, we also notice that although the plot is symmetric for p = .**5, it is not**
for both p = .1 and p = .9, i.e. absolute changes of ∆p and −∆p **give us different amounts**
of log-odds change. For example, changing the parameter from .1 to .**05 give us a larger**
log-odds change than changing the parameter from .1 to .**15. We also notice that the plots** for p = .1 and p = .**9 are mirror images of each other. Therefore, the log-odds change is the**
same for complementary parameter changes on θx|u and θx|u.

We close this section by emphasizing that the above figures identify parameter changes that guarantee keeping queries within certain ranges. However, if the belief network has specific properties, such as a specific topology, then it is possible for the query to be robust against parameter changes that are outside the identified bounds.

## 5. Conclusion

In this paper, we presented an efficient technique for fine-tuning **the parameters of a belief** network. The technique suggests minimal changes to network **parameters which ensure**
that certain constraints are enforced on probabilistic queries. Based on this technique, we have experimented with some belief networks, only to find out that these networks are more sensitive to parameter changes than previous experimental studies seem to suggest.

This observation leads us to an analytic study on the effect of parameter changes, with the aim of characterizing situations under which parameter changes do or do not matter. We have reported on a number of results in this direction. Our central result shows that belief networks are robust in a very specific sense: the relative change in query odds is bounded by the relative change in the parameter odds. A closer look at **this result, its meaning,** and its implications provides interesting characterizations of parameter changes that do or do not matter, and explains analytically some of the previous experimental results and observations on this matter.

## Acknowledgments

A shorter version of this paper appeared in Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence (UAI-01)**, pp. 65–74. This work has been partially supported**
by NSF grant IIS-9988543, MURI grant N00014-00-1-0617, and by DiMI grant 00-10065.

## Appendix A. Proofs

Theorem 2.1 The derivative of Pr (e) with respect to the meta parameter τx|u **is given**
by:

$${\frac{\partial P r(\mathbf{e})}{\partial\tau_{x|\mathbf{u}}}}={\frac{P r(\mathbf{e},x,\mathbf{u})}{\theta_{x|\mathbf{u}}}}-{\frac{P r(\mathbf{e},{\overline{{x}}},\mathbf{u})}{\theta_{{\overline{{x}}}|\mathbf{u}}}},$$

when θx|u 6= 0 and θx|u 6= 0.