Existence theorem of the quadruple (P, R, F, M): Precision, recall, fallout and miss

In an earlier paper [Egghe, L. (2004). A universal method of information retrieval evaluation: the “missing” link M and the universal IR surface. Information Processing and Management, 40, 21–30] 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 MP1-P1-RRF1-F1-MM=1In 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.