BM25: The Relevance Formula Behind Almost Every Search Box

The Question BM25 Answers

You type climate policy. Ten thousand documents contain some of those words. Which one goes first? BM25 (Best Matching 25) is the standard answer — a probabilistic scoring function from the 1990s that remains the default in Lucene, Elasticsearch, OpenSearch, and most modern engines as a first-stage ranker, even when a learning model reranks later.

The Three Ingredients

For each query term, BM25 combines:

• TF — term frequency, with saturation. A document mentioning the term more often scores higher — but with diminishing returns. The jump from 1 to 5 occurrences matters; from 50 to 54 barely registers. The k1 parameter controls how fast the curve flattens (typically ~1.2–2.0). This saturation is why keyword-stuffing stopped working decades ago.

• IDF — inverse document frequency. Rare words weigh more than common ones. If the appears everywhere, matching it proves nothing; if quorum appears in three documents, matching it is strong evidence. This is the "rare term = informative term" principle in one multiplication.

• B — length normalisation. Long documents naturally contain more query terms by accident. The b parameter (typically 0.75) divides the score by a length factor so a short, focused page can beat an encyclopedia entry that merely mentions the topic in passing.

$$
\text{score}(q,d) = \sum_{t \in q} \text{IDF}(t) \cdot \frac{f(t,d)\,(k_1+1)}{f(t,d) + k_1\left(1-b+b\cdot\frac{|d|}{\text{avgdl}}\right)}
$$

You do not need to memorise the algebra. You need the three knobs: frequency saturates, rare words count more, long docs are normalised.

Where BM25 Beats Cleverness

• It is explainable. For any result you can print: which terms matched, their frequencies, their IDF. Embeddings hand you a number and a shrug.

• It is instant. Inverted-index lookup plus arithmetic — no model inference, no GPU, milliseconds on hundreds of thousands of documents.

• It fails honestly. Zero matching terms → low score → the engine can say it found nothing. A neural model will still produce a confident-looking answer with nothing behind it.

• Exact matches win when they should. Searching a product code, a hostname, or a surname rewards literal matching, where BM25 is strongest.

Where It Stops

BM25 does not know that car and automobile are the same thing, cannot follow "the president of France" across paraphrase, and has no notion of intent. That is where synonyms, stemming, and neural reranking enter — typically on top of a BM25 first stage that has already cut a million documents to the top hundred.

Practical Takeaways

• Tuning k1 and b is real but secondary — fix your data first: clean titles, de-duplicated pages, sensible field lengths. Garbage in, formula out.

• Search across fields with weights (title > headings > body) before touching parameters.

• Measure with real queries: collect actual searches (and zero-result ones), score candidates by hand for twenty queries, and change one variable at a time.

Every search engine you admire almost certainly runs BM25 somewhere in its pipeline. Understanding it is the difference between tuning your engine and merely turning knobs.