BM25, Best Matching 25, is a probabilistic ranking function used in information retrieval to score how relevant a document is to a query. It is a refined version of TF IDF and is widely used in search engines and classical retrieval systems

Before reading this and applying this you need to know about TF-IDF and Inverted Index.

Why it exists?

IDF Problem

IDF is calculated as log(N + 1 / df + 1)

• N = total number of documents in the collection
• df = document frequency (how many documents contain this term)
• log = logarithm function (reduces the impact of very large numbers)

The problems with this formula:

• Division by zero: When df = 0 (we "solved" this by adding 1 to both the numerator and denominator)
• Unstable rare terms: Very rare terms get extremely high scores
• Negative scores: Very common terms can get negative scores

BM25 solved this by comparing the documents that don’t contain the term against documents that do contain it:

$$ IDF(t) = \log \frac{N - df + 0.5}{(df + 0.5) + 1} $$

• Top part (N - df + 0.5): How many documents DON'T have this word (with a small adjustment)
• Bottom part (df + 0.5): How many documents DO have this word (with a small adjustment)
• Why add 0.5? To avoid dividing by zero. This technique is called Laplace smoothing.
• Why add 1? The + 1 at the end makes sure the score never goes negative. This helps handle unusual cases.

Term Frequency Saturation