2. or increase the prior probability of tampering from its current value of .02 to ≥ .**036.**
Therefore, the distinctions between .02 and .036, and the one between .01 and .**0047, do**
really matter in this case as each induces an absolute change of .**15 on the probabilistic query**
of interest. Note also that implicit in SamIam**'s recommendations is that the parameters**
of variables Fire, Smoke, Leaving, and Alarm **are irrelevant to enforcing this constraint, i.e.** no matter how much we change any of these parameters, we would **not be able to enforce** our desired constraint.

This example shows that the absolute change **in a query can be much larger than the**
absolute change **in the corresponding parameters. Later, we will show an example where**
an infinitesimal change to a network parameter leads to a change of .**5 to a corresponding**
query. We also show examples in which the relative change **in the probability of a query is**
larger than the corresponding relative change **in a network parameter. One wonders then** whether there is a different method for measuring probabilistic change (other than absolute or relative), which allows one to non-trivially bound the change in a probabilistic query in terms of the corresponding change in a network parameter.

To answer this and related questions, we conduct in Section 3 **an analytic study of the**
partial derivative of a probabilistic query Pr (y | e**) with respect to some network parameter**
θx|u**. Our study leads us to three main results:**
1. a bound on the derivative in terms of Pr (y | e) and Pr (x | u**) only, which is independent of any other aspect of the given belief network;**
2. a bound on the sensitivity of queries to infinitesimal changes in network parameters; 3. a bound on the sensitivity of queries to arbitrary changes **in network parameters.**
The last bound in particular shows that the amount of change in a probabilistic query can be bounded in terms of the amount of change in a network parameter, as long as change is understood to be the relative change in odds**. This result has a number of practical implications. First, it can relieve experts from having to be too precise when specifying certain**
parameters subjectively. Next, it can be important for approximate inference algorithms that pre-process network parameters to eliminate small distinctions in such parameters, in order to increase the efficiency of inference (Poole, 1998). Finally, it can be used to show that automated reasoning systems based on belief networks are robust and, hence, suitable for real-world applications (Pradhan, Henrion, Provan, Del **Favero, & Huang, 1996).**
Section 4 is indeed dedicated to exploring the implications **of the above bounds, where**
we provide an analytic explanation of why certain parameter **changes don't matter. We** finally close in Section 5 with some concluding remarks. Proofs of all theorems are given in Appendix A.

## 2. The Tuning Of Network Parameters

We report in this section on a tool that we have been developing, called SamIam**, for finetuning network parameters (Laskey, 1995; Castillo, Guti´errez, & Hadi, 1997; Jensen, 1999;**
Kjærulff & van der Gaag, 2000; Darwiche, 2000). Given a belief network, some evidence e, which is an instantiation of variables E in the belief network, and two events y and z of variables Y and Z respectively, where Y, Z 6∈ E**, our tool can efficiently identify parameter**
changes needed to enforce the following types of constraints: