BM25

BM25 (Best Matching 25)

BM25 is a probabilistic ranking function based on the binary independence model. It is the most widely-used unsupervised retrieval model and the standard baseline in information retrieval.

BM25 Scoring Function

$\text{BM25}(q,d)={\sum }_{t\in q}\text{IDF}(t)\cdot \frac{f(t,d)\cdot (k_1+1)}{f(t,d)+k_1\cdot \left(1-b+b\cdot \frac{|d|}{\text{avgdl}}\right)}$

where:

• $f(t,d)$ — term frequency of term $t$ in document $d$
• $|d|$ — length of document $d$ (in words)
• $\text{avgdl}$ — average document length in the collection
• $k_1$ — term frequency saturation parameter (typically 1.2-2.0)
• $b$ — length normalization parameter (typically 0.75)
• $\text{IDF}(t)=\log \frac{N-n(t)+0.5}{n(t)+0.5}$ — inverse document frequency

What Each Part Does

• IDF: Rare terms are more informative → higher weight
• TF saturation: First occurrence matters most; additional occurrences have diminishing returns (controlled by $k_1$). As $k_1\to \infty$, approaches raw TF; as $k_1\to 0$, binary presence/absence.
• Length normalization: Longer documents naturally have higher TF. Parameter $b$ controls how much to penalize length. $b=1$: full normalization; $b=0$: no normalization.

Parameter Effects

Parameter	Low	High
$k_1$	Binary (term present/absent)	Raw term frequency
$b$	No length normalization	Full length normalization

Connections

• Derived from: Binary Independence Model, probabilistic retrieval
• Compared with: TF-IDF, Query Likelihood Model
• Neural extensions: provides first-stage retrieval for Neural Reranking, Multi-Stage Ranking
• Sparse baselines: Dense Retrieval and Learned Sparse Retrieval are often benchmarked against BM25

Appears In

• IR-L03 - Retrieval Models
• IR-A01 - Unsupervised Retrieval