Existence theorem of the quadruple (P, R, F, M): Precision, Recall, Fallout and Miss

Abstract

In an earlier paper [L. Egghe. A universal method of information retrieval evaluation: the “missing “ link M and the universal IR surface, Information Processing and Management 40, 21-30, 2004] we showed that, given an IR system, and if P denotes precision, R recall, F fallout and M miss (re-introduced in the paper mentioned above), we have the following relationship between P, R, F and M: P/1-P * 1-R/R* F/1-F * 1-M/M = 1.

In this paper we prove the (more difficult) converse: given any four rational numbers in the interval ]0,1[ satisfying the above equation, then there exists an IR system such that these four numbers (in any order) are the precision, recall, fallout and miss of this IR system. As a consequence we show that any three rational numbers in ]0,1[ represent any three measures taken from precision, recall, fallout and miss of a certain IR system. We also show that this result is also true for two numbers instead of three.