A Geometric View of Relevance Effectiveness in Information Retrieval

Relevance is a central concept in Information Retrieval (IR). It is used to work out effectiveness measures for IR systems, i.e. measures to express how well (or bad) an IR system performs; classical measures are precision, recall, fallout. It is shown that the empirical relation P=NR/x (P=precision, R=recall, N=total number of relevant documents, x=the number of retrieved documents) can be formally easily obtained. It is also shown that using the concept of fallout a typical surface can be constructed with the noteworthy properties that it looks similarly for every IR system and each point on this surface corresponds to a 3-tuple (precision, recall, fallout) and thus to one retrieval process. Thus, the name of effectiveness surface is suggested for it. The performance of an IR system can be enhanced by a technique called relevance feedback (used to return documents that are likely to be more relevant). A sequence of repeatedly applied relevance feedbacks, being a sequence of repeated retrievals, corresponds to a sequence of points (‘walk’) on the effectiveness surface. It is shown that this sequence can be theoretically modelled by an important mathematical structure (recursively enumerable set or Diophantine set), and that it yields a point on the effectiveness surface corresponding to an optimal retrieval situation. Further, the existence of an optimal point is also shown and it is computed as well.