Binary Independence Model
The Binary Independence Model (BIM) in computing and information science is a probabilistic information retrieval technique. The model makes some simple assumptions to make the estimation of document/query similarity probable and feasible. Definitions. The Binary Independence Assumption is that documents are binary vectors. That is, only the presence or absence of terms in documents are recorded. Terms are independently distributed in the set of relevant documents and they are also independently distributed in the set of irrelevant documents. The representation is an ordered set of Boolean variables. That is, the representation of a document or query  is a vector with one Boolean element for each term under consideration. More specifically, a document is represented by a vector where if term "t" is present in the document "d" and if it's not. Many documents can have the same vector representation with this simplification. Queries are represented in a similar way. "Independence" signifies that terms in the document are considered independently from each other and no association between terms is modeled. This assumption is very limiting, but it has been shown that it gives good enough results for many situations. This independence is the "naive" assumption of  a Naive Bayes classifier, where properties that imply each other are nonetheless treated as independent for the sake of simplicity. This assumption allows the representation to be treated as an instance of a Vector space model by considering each term as a value of 0 or 1 along a dimension orthogonal to the dimensions used for the other terms. The probability formula_1 that a document is relevant derives from the probability of relevance of the terms vector of that document formula_2. By using the Bayes rule we get: where formula_4 and formula_5 are the probabilities of retrieving a relevant or  nonrelevant document, respectively. If so, then that document's representation is "x". The exact probabilities can not be known beforehand, so estimates from statistics about the collection of documents must be used. formula_6 and formula_7 indicate the previous probability of retrieving a relevant or nonrelevant document respectively for a query "q". If, for instance, we knew the percentage of relevant documents in the collection, then we could use it to estimate these probabilities. Since a document is either relevant or nonrelevant to a query we have that: Query Terms Weighting. Given a binary query and the dot product as the  similarity function between a document and a query, the problem is to assign weights to the terms in the query such that the retrieval effectiveness will be high. Let formula_9 and formula_10 be the probability that a relevant document and an irrelevant document has the term respectively. Yu and Salton, who first introduce BIM, propose that the weight of the term is an increasing function of formula_11. Thus, if formula_12 is higher than formula_13, the weight of term will be higher than that of term . Yu and Salton showed that such a weight assignment to query terms yields better retrieval  effectiveness than if query terms are equally weighted. Robertson and Spärck Jones later showed that if the term is assigned the weight of formula_14, then optimal retrieval effectiveness is obtained under the Binary Independence Assumption. The Binary Independence Model was introduced by Yu and Salton. The name Binary Independence Model was coined by Robertson and Spärck Jones.