Interpretations of TF-IDF are based on binary independence retrieval, Poisson, information theory, and language modelling. This paper contributes a review of existing interpretations, and then, TF-IDF is systematically related to the probabilities P(q|d) and P(d|q). Two approaches are explored: a space of independent, and a space of disjoint terms. For independent terms, an "extreme" query/non-query term assumption uncovers TF-IDF, and an analogy of P(d|q) and the probabilistic odds O(r|d, q) mirrors relevance feedback. For disjoint terms, a relationship between probability theory and TF-IDF is established through the integral + 1/x dx = log x. This study uncovers components such as divergence from randomness and pivoted document length to be inherent parts of a document-query independence (DQI) measure, and interestingly, an integral of the DQI over the term occurrence probability leads to TF-IDF.

This paper investigates in a stringent athematical formalism the parallel derivation of three grand probabilistic retrieval models: binary independent retrieval (BIR), Poisson model (PM), and language modelling (LM).The investigation has been motivated by a number of questions. Firstly, though sharing the same origin, namely the probability of relevance, the models differ with respect to event spaces. How can this be captured in a consistent notation, and can we relate the event spaces? Secondly, BIR and PM are closely related, but how does LM fit in? Thirdly, how are tf-idf and probabilistic models related? .The parallel investigation of the models leads to a number of formalised results: