We consider in this chapter the relationship between the values of parameters that quantify a Bayesian network and the values of probabilistic queries applied to these networks. In particular, we consider the impact of parameter changes on query values, and the amount of parameter change needed to enforce some constraints on these values.

Introduction

Consider a laboratory that administers three tests for detecting pregnancy: a blood test, a urine test, and a scanning test. Assume also that these tests relate to the state of pregnancy as given by the network of Figure 16.1 (we treated this network in Chapter 5). According to this network, the prior probability of pregnancy is 87% after an artificial insemination procedure. Moreover, the posterior probability of pregnancy given three negative tests is 10.21%. Suppose now that this level of accuracy is not acceptable: the laboratory is interested in improving the tests so the posterior probability is no greater than 5% given three negative tests. The problem now becomes one of finding a certain set of network parameters (corresponding to the tests' false positive and negative rates) that guarantee the required accuracy. This is a classic problem of sensitivity analysis that we address in Section 16.3 as it is concerned with controlling network parameters to enforce some constraints on the queries of interest.

Assume now that we replace one of the tests with a more accurate one, leading to a new Bayesian network that results from updating the parameters corresponding to that test.