BM25 is a bag-of-words retrieval function that ranks a set of documents based on the query terms appearing in each document, regardless of their proximity within the document. It is a family of scoring functions with slightly different components and parameters. One of the most prominent instantiations of the function is as follows. Given a query, containing keywords, the BM25 score of a document is: where is 's term frequency in the document, is the length of the document in words, and is the average document length in the text collection from which documents are drawn. and are free parameters, usually chosen, in absence of an advanced optimization, as and. is the IDF weight of the query term. It is usually computed as: where is the total number of documents in the collection, and is the number of documents containing. There are several interpretations for IDF and slight variations on its formula. In the original BM25 derivation, the IDF component is derived from the Binary Independence Model. The above formula for IDF has drawbacks for terms appearing in more than half of the corpus documents. These terms' IDF is negative, so for any two almost-identical documents, one which contains the term may be ranked lower than one which does not. This is often an undesirable behavior, so many applications adjust the IDF formula in various ways:
Each summand can be given a floor of 0, to trim out common terms;
The IDF function can be given a floor of a constant, to avoid common terms being ignored at all;
The IDF function can be replaced with a similarly shaped one which is non-negative, or strictly positive to avoid terms being ignored at all.
Here is an interpretation from information theory. Suppose a query term appears in documents. Then a randomly picked document will contain the term with probability . Therefore, the information content of the message " contains " is: Now suppose we have two query terms and. If the two terms occur in documents entirely independently of each other, then the probability of seeing both and in a randomly picked document is: and the information content of such an event is: With a small variation, this is exactly what is expressed by the IDF component of BM25.
Modifications
At the extreme values of the coefficient BM25 turns into ranking functions known as BM11 and BM15.
BM25F is a modification of BM25 in which the document is considered to be composed from several fields with possibly different degrees of importance, term relevance saturation and length normalization.
BM25+ is an extension of BM25. BM25+ was developed to address one deficiency of the standard BM25 in which the component of term frequency normalization by document length is not properly lower-bounded; as a result of this deficiency, long documents which do match the query term can often be scored unfairly by BM25 as having a similar relevancy to shorter documents that do not contain the query term at all. The scoring formula of BM25+ only has one additional free parameter as compared with BM25:
