BM25+

What it is

BM25+ is a modification of BM25 that addresses a known flaw: in the original BM25, a term’s TF contribution can approach zero for very long documents due to the length normalisation penalty. This means a very long document that mentions a query term once might score lower than a short document that doesn’t mention the term at all — a counterintuitive result called the lower-bounding problem.

BM25+ adds a small constant δ (delta) to the TF component to ensure that any document containing a term receives a minimum positive contribution from that term.

How it works

Standard BM25 term weight:

tf_bm25(t, d) = (tf · (k₁ + 1)) / (tf + k₁ · (1 - b + b · |d| / avgdl))

For a very long document (|d| >> avgdl) with tf=1:

denominator ≈ k₁ · b · (|d| / avgdl)  →  very large
→  tf_bm25 → 0

BM25+ formula:

tf_bm25plus(t, d) = δ + (tf · (k₁ + 1)) / (tf + k₁ · (1 - b + b · |d| / avgdl))

The δ parameter (typically 1.0) guarantees the TF component never falls to zero. A document containing a term always scores strictly higher than one not containing it — regardless of document length.

[illustrate: BM25 vs BM25+ TF contribution curves for a fixed k₁ and b — x-axis is document length (1× to 20× avgdl), y-axis is TF weight for a single occurrence — BM25 curve descending toward zero for long documents, BM25+ curve asymptoting to δ (dashed floor) rather than zero]

Example

Three documents for query "rare_term":

Under BM25, D3 ranks below D1 (which doesn’t contain the term at all) — clearly wrong. BM25+ ensures D2 and D3 both score higher than D1, with D2 still scoring higher due to its shorter length.

Variants and history

BM25+ was proposed by Lv and Zhai in their 2011 paper “Lower-Bounding Term Frequency Normalization.” The paper demonstrated the flaw empirically on TREC collections and showed that adding δ = 1 consistently improved ranking quality.

BM25L is a related variant from the same paper that addresses the same problem using a different approach — capping the length normalisation factor rather than adding a floor.

When to use it

BM25+ is most beneficial when your corpus contains a wide range of document lengths — for example, mixing short product descriptions with long user manuals in the same index. In practice, BM25 and BM25+ perform similarly on homogeneous corpora.

Native BM25+ is not available as a built-in option in Elasticsearch or OpenSearch as of writing — the default BM25 implementation does not include the δ parameter. Custom Lucene Similarity implementations can add it. Some newer search libraries (Tantivy, Meilisearch) expose BM25 parameter tuning that may be extended to BM25+.

See also

• BM25
• Okapi BM25
• BM25F
• TF-IDF
• Term Frequency