The Book of Why: The New Science of Cause and Effect

by Judea Pearl

We really mean that any prior causes of A can be adequately summarized in the prior probability P(A) that A is true. For example, in the Disease Test example, family history might be a cause of Disease. But as long as we are sure that this family history will not affect the variable Test (once we know the status of Disease), we need not represent it as a node in the graph. However, if there is a cause of Disease that also directly affects Test, then that cause must be represented explicitly in the diagram. In the case where the node A has a parent, A has to “listen” to its parent before deciding on its own state. In our mammogram example, the parent of Test was Disease. We can show this “listening” process with a 2 × 2 table (see Table 3.2). For example, if Test “hears” that D = 0, then 88 percent of the time it will take the value T = 0, and 12 percent of the time it will take the value T = 1. Notice that the second column of this table contains the same information we saw earlier from the Breast Cancer Surveillance Consortium: the false positive rate (upper right corner) is 12 percent, and the sensitivity (lower right corner) is 73 percent. The remaining two entries are filled in to make each row sum to 100 percent. TABLE 3.2. A simple conditional probability table.