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Equation 8 · Part 6 · From BM25 to Agentic Retrieval: A History of Retrieval-Augmented Generation

Symbol b

score(D,Q)=∑qi∈QIDF(qi)⋅f(qi,D)⋅(k1+1)f(qi,D)+k1⋅(1−b+b⋅∣D∣avgdl),\mathrm{score}(D, Q) = \sum_{q_i \in Q} \mathrm{IDF}(q_i) \cdot \frac{f(q_i, D) \cdot (k_1 + 1)}{f(q_i, D) + k_1 \cdot \left(1 - b + b \cdot \dfrac{|D|}{\mathrm{avgdl}}\right)},
bb

What this part means

tuned constants controlling term-frequency saturation and length normalization respectively.

Its job in the formula

b occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Where the article explains it

where f(qiq_i, D) is the frequency of query term qiq_i in D , |D| is the document’s length, avgdl\mathrm{avgdl} is the average document length in the collection, and k1k_1 and b are tuned constants controlling term-frequency saturation and length normalization respectively.

The passage around this formula

…D scores against a query Q as score(D,Q)=∑qi∈QIDF(qi)⋅f(qi,D)⋅(k1+1)f(qi,D)+k1⋅(1−b+b⋅∣D∣avgdl)\mathrm{score}(D, Q) = \sum_{q_i \in Q} \mathrm{IDF}(q_i) \cdot \frac{f(q_i, D) \cdot (k_1 + 1)}{f(q_i, D) + k_1 \cdot \left(1 - b + b \cdot \dfrac{|D|}{\mathrm{avgdl}}\right)}. where f(qiq_i, D) is the frequency of query term qiq_i in D , |D| is the document’s length, avgdl\mathrm{avgdl} is the average document length in the collection, and k1k_1 and b are tuned constants controlling term-frequency saturation and length normalization respectively. The saturation term is the substantive advance over a raw term-frequency-times-IDF score: a term’s contribution grows quickly at first and then flattens, so a document that happens to repeat a query…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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