Rule inference

A rule can be defined as a statement ``if event $\mathcal{X}$ occurs, then event $\mathcal{Y}$ is likely to occur'', where the events are propositions of the form of variables taking particular values from their state sets (adapted from [41]). In other words events $\mathcal{X}$ and $\mathcal{Y}$ can be described as sets of (variable, value)-pairs; the purpose is to find these sets $\mathcal{X}$ and $\mathcal{Y}$ such that $\mathcal{X}$ ``implies'' (with high probability) $\mathcal{Y}$, denoted as $(\mathcal{X} \rightarrow \mathcal{Y})$. We need to be careful with this notation as a rule does not necessarily imply causality (Section 4).

We already know a kind of rule induction from the classification problem. There we fix $\mathcal{Y}$ as the classification variable with one of its values and we try to find sets of $\mathcal{X}$ which are good predictors for the right classification. As we ``supervise'' this process with specific classification data it is called a supervised method. In `general rule finding' we look for regions of high structure anywhere in the relation to obtain a better understanding of the domain (expert knowledge). This is well summarized in Smyth and Goodman [41, pg. 302, 313]: ``Classification only derives rules relating to a single `class' attribute, whereas generalized rule induction derives rules relating any or all of the attributes''. ``The rules produced (...) can be used either as a human aid to understanding the inherent model embodied in data, or as a tentative input set of rules to an expert system.''

General rule induction is therefore defined as an unsupervised method, even though this distinction seems somewhat `fuzzy'. Also in general rule induction we sometimes like to specify parts of the event-sets $\mathcal{X}$ or $\mathcal{Y}$a priori, though not directly for classification purposes but for approaching the problem in a more user guided manner For simplicity we only deal with $\mathcal{Y}$ containing one (variable, value)-pair. Rules with more ``implications'' can always be divided in several rules with one implication.